1 Introduction

By the Atlas of finite groups [18], one can see that the symplectic groups Sp(8, 2) has 11 conugacy classes of maximal subgroups. The fourth maximal subgroup of Sp(8, 2) is a group of the form \(2^{10}{:}A_{8}:= \overline{G}.\) This is a split extension group of the elementary abelian group \(2^{10}\) of order 1024, by the alterntain group \(A_{8}.\) In this article we focus on the group \(\overline{G},\) where we will determine its conjugacy classes, the inertia factors of this extension with the fusions of their conjugacy classes into the classes of \(A_{8},\) the character tables of these inertia factors and finally the full character table of the full extension \(\overline{G}.\) We used the coset analysis method together with the Clifford–Fischer theory to establish the conjugacy classes and the character table of \(\overline{G}\). The most interesting part is the determination of the inertia factor groups, where there are 7 inertia factor groups, namely \(H_{1} = A_{8}\), \(H_{2},\ldots , H_{7}.\) The main method used to determine the structures of \(H_{2},\ldots , H_{7}\) is by analysing the maximal subgroups of \(A_{8}\) and the maximal subgroups of these maximal subgroups. Sometimes we consider the third level of maximal subgroups of \(A_{8}.\) The determination of \(H_{2},\) \(H_{3}\) and \(H_{5}\) was straightforward from the maximal subgroups of \(A_{8}.\) There are many possibilities for \(H_{4},\) \(H_{6}\) and \(H_{7}\) and all possibilities lead to contradictions (using various information from Clifford–Fischer theory and the interplay between the conjugacy classes of \(\overline{G}\) obtained using the coset analysis technique and the Fischer matrices); except one possibility, where we ended up with finding that \(H_{2} = 2^{3}{:}GL(3,2),\) \(H_{3} = 2^{4}{:}(S_{3}\times S_{3}),\) \(H_{4} = 2^{3}{:}S_{4},\) \(H_{5} = S_{5},\) \(H_{6} = (S_{3}\times S_{3}){:}2\) and \(H_{7} = 2 \times S_{4}.\) The Fischer matrices of \(\overline{G}\) have all been determined in this paper and their sizes range between 1 and 16. The character table of \(\overline{G}\) is a \(81 \times 81\) \(\mathbb {C}\)-valued matrix and it is partitioned into 98 parts corresponding to the 7 inertia factor groups and the 14 conjugacy classes of \(G = A_{8}.\) If one was only interested in the calculation of the character table of the group \(\overline{G}\), then it could be computed by using GAP [25] or Magma [15] and the generators \(\overline{g}_{1}\) and \(\overline{g}_{2}\) of \(\overline{G},\) given below. But Clifford–Fischer Theory provides much more interesting and practical information on the group and on the character table, in particular the character table produced by Clifford–Fischer Theory is in a special format that could not be achieved by direct computations using GAP or Magma. Also providing examples of applications of Clifford–Fischer Theory to both split and non-split extensions is sensible choice, since each group requires individual approach. The readers (particularly young researchers) will highly benefit from the theoretical background required for these computations. GAP and Magma are computational tools and would not replace good powerful and theoretical arguments. In addition, there is an interesting interplay between the coset analysis and Clifford–Fischer Theory. Indeed the size of each Fischer matrix is \(c(g_{i}),\) the number of \(\overline{G}\)-classes corresponding to \([g_{i}]_{G}\) obtained via the coset analysis technique. That is computations of the conjugacy classes of \(\overline{G}\) using the coset analysis technique will determine the sizes of all Fischer matrices.

By the electronic Atlas of Wilson [27], we can see that Sp(8, 2) acts on 120 points and therefore it can generated in terms of 120 points. With few GAP commands we were able to construct our split extension group \(\overline{G} = 2^{10}{:}A_{8}\) in terms of the set \(\{1,2,\ldots ,120\}.\) The following two elements \(\overline{g}_{1}\) and \(\overline{g}_{2}\) generate \(\overline{G}.\)

$$\begin{aligned} \overline{g}_{1}= & {} (1,113,20,115,100,105,101)(2,77,10,86,92,94,28)\\&(3,114,78,87,91,97,49,24,48,25,66,\\&57,95,56)(4,111,52,39,5,16, 68,64,37)(33,46,108,84,63,102,62)\\&(38,104,59,83,106,79,\\&71) 69,34,21,40,53,11,7,27)(6,88,60,93,22,72,36)\\&(8,112,13,74,67, 117,45)(12,90,118,47,17,55,110)\\&(14,35,107,26,23,41,116)(15,19,18,58,50,61,81,99,51,70,76,73,44,82)\\&(29,32,103,98,30,54,85)(31,80,65,109,120,89,43,96,75,119,42, 68,64,37)\\&(33,46,108,84,63,102,62)(38,104,59,83,106,79,71),\\ \overline{g}_{2}= & {} (1,35,95,60,83,31,14,93,58,46,54,34)\\&(2,15,75,119,67,16,86,68,80,44,12,51)(3,101,7,\\&10,5,50,40,33,19,92,25, 73)(4,30,108,76,59,118,96,117,113,97,90,45)\\&(6,17,85,81,69, 13,103,79,107,114,18,55)\\&(8,48,27,100,62,72,20,82,70,102,74,63)\\&(9,99,21,24,116,66,\\&77,120,111,52,88,89)(11,49,61,104,64,94,78,56,65,98,87,28)\\&(22,53,57,32,38,29,106,\\&109,37,71,41,112)(23,105,26,47,39,43,36,110,115,84,42,91), \end{aligned}$$

where \(o(\overline{g}_{1}) = 14,\) \(o(\overline{g}_{2}) = 10\) and \(o(\overline{g}_{1}\overline{g}_{2}) = 24.\)

Having \(\overline{G}\) being constructed in GAP, it is easy to obtain all its normal subgroups. In fact \(\overline{G}\) possesses two proper normal subgroups of orders 64 and 1024. The normal subgroup of order 1024 is an elementary abelian group isomorphic to N. In GAP one can check for the complements of N in \(\overline{G}\), where in our case we obtained two complements both isomorphic to \(A_{8}\) and any of these two complements together with N gives the split extension in consideration.

It is worth mentioning that the group in question was also treated by C. Chileshe in his PhD thesis [16], but in a different way. One of the major differences between the two approaches is that Chileshe developed his own GAP routines (or adapted other existing routines) to do most of his computations where he worked with \(\overline{G}\) as a matrix group, while we generated our group in terms of permutations to perform our calculations. Also the way that we determined the inertia factor groups is totally different where we investigated maximal subgroups of the maximal subgroups of \(A_{8}.\)

For the notation used in this paper and the description of Clifford–Fischer theory technique, we follow [1,2,3,4,5,6,7,8,9,10,11,12,13,14, 17, 19].

2 Conjugacy classes of \(\overline{G} = 2^{10}{:}A_{8}\)

In this section we compute the conjugacy classes of the group \(\overline{G}\) using the coset analysis technique (see Basheer [2], Basheer and Moori [3, 4, 6] or Moori [21] and [22] for more details) as we are interested to organize the classes of \(\overline{G}\) corresponding to the classes of \(A_{8}.\) Firstly note that \(A_{8}\) has 14 conjugacy classes (see the Atlas). Corresponding to these 14 classes of \(A_{8},\) we obtained 81 classes in \(\overline{G}.\)

In Table 1, we list the conjugacy classes of \(\overline{G}\), where in this table:

  • \(k_{i}\) is the number of orbits \(Q_{i1}, Q_{i2},\ldots ,Q_{ik_{i}}\) for the action of N on the coset \(N\overline{g}_{i} = Ng_{i}\), where \(g_{i}\) is a representative of a class of the complement (\(A_{8}\)) of N in \(\overline{G}\). In particular, the action of N on the identity coset N produces 1024 orbits each consists of singleton. Thus for \(\overline{G},\) we have \(k_{1}= 1024.\)

  • \(f_{ij}\) is the number of orbits fused together under the action of \(C_{G}(g_{i})\) on \(Q_{1}, Q_{2},\ldots ,Q_{k}.\) In particular, the action of \(C_{G}(1_{G})\) \(= G\) on the orbits \(Q_{1}, Q_{2},\ldots ,Q_{1024}\) affords 7 orbits of lengths 1, 15, 28, 35, 105 and 420 (twice) (with corresponding point stabilizers \(A_{8},\) \(2^{3}{:}GL(3,2),\) \(S_{6},\) \(2^{4}{:}(S_{3} \times S_{3}),\) \(2^{3}{:}S_{4}\) and \(2\times S_{4}\) (twice). Thus \(f_{11} = 1,\) \(f_{12} = 15,\) \(f_{13} = 28,\) \(f_{14} = 35,\) \(f_{15} = 105,\) and \(f_{16} = f_{17} = 420.\)

  • \(m_{ij}\)’s are weights (attached to each class of \(\overline{G}\)) that will be used later in computing the Fischer matrices of \(\overline{G}.\) These weights are computed through the formula

    $$\begin{aligned} m_{ij} = [N_{\overline{G}}(N\overline{g}_{i}):C_{\overline{G}}(g_{ij})] = |N|\frac{|C_{G}(g_{i})|}{|C_{\overline{G}}(g_{ij})|}, \end{aligned}$$
    (1)

    where N is the kernel of an extension \(\overline{G}\) that is in consideration.

Table 1 The conjugacy classes of \(\overline{G} = 2^{10}{:}A_{8}\)
Full size table

3 Inertia factor groups of \(\overline{G} = 2^{10}{:}A_{8}\)

We have seen in Sect. 2 that the action of \(\overline{G}\) on N produced 7 orbits of lengths 1, 15, 28, 35, 105 and 420 (twice). By a theorem of Brauer (for example see Theorem 5.1.1 of Basheer [2]), it follows that the action of \(\overline{G}\) (or just G) on \(\mathrm {Irr}(N)\) will also produce 7 orbits and thus there will be 7 inertia factor groups \(H_{1}, H_{2}, \ldots , H_{7}.\) With the help of GAP, we found that the lengths of the orbits on the action of G on \(\mathrm {Irr}(N)\) are 1, 15, 35, 105, 168, 280 and 420. Therefore the indices of \(H_{1}, H_{2}, \ldots , H_{7}\) in \(G= A_{8}\) are 1, 15, 35, 105, 168, 280 and 420 respectively. By looking at the maximal subgroups of \(A_{8},\) that available in the Atlas [18], it is readily verified that \(H_{1} = A_{8},\) \(H_{2} = 2^{3}{:}GL(3,2)\) and \(H_{3} = 2^{4}{:}(S_{3} \times S_{3}).\) In Tables 2 and 3, we supply the full character tables of the inertia factor groups \(H_{2}\) and \(H_{3}\) together with the fusions of their conjugacy classes into the classes of \(A_{8}.\)

Table 2 The character table of \(H_{2} = 2^{3}{:}GL(3,2)\)
Full size table
Table 3 The character table of \(H_{3} = 2^{4}{:}(S_{3}\times S_{3})\)
Full size table

Next we determine the structures of \(H_{4},\) \(H_{5},\) \(H_{6}\) and \(H_{7}\) by analysing the maximal subgroups of \(A_{8}.\) For \(H_{4},\) we can see from the maximal subgroups of \(A_{8},\) that available in the Atlas, that \(H_{4}\) is either:

  • an index 7 subgroup of \(2^{3}{:}GL(3,2)\) or

  • an index 3 subgroup of \(2^{4}{:}(S_{3} \times S_{3}).\)

The group \(2^{3}{:}GL(3,2)\) has 5 conjugacy classes of maximal subgroups, where representatives of these classes have orders 192 (twice) and 168 (three times) with indices 7 (twice) and 8 (three times) respectively. The two groups of order 192 are of the form \(2^{3}{:}S_{4}\) (two non-isomorphic groups), where one group has 14 conjugacy classes of elements while the second one has 13 conjugacy classes of elements. Thus if \(H_{4} \le 2^{3}{:}GL(3,2)\) such that \([(2^{3}{:}GL(3,2)):H_{4}] = 7,\) then \(H_{4}\) is either \(2^{3}{:}S_{4}\) with 14 irreducible characters, or \(H_{4}\) is \(2^{3}{:}S_{4}\) with 13 irreducible characters. We look at the other possibility, where we assume that \(H_{4}\) is an index 3 subgroup of \(2^{4}{:}(S_{3} \times S_{3}).\) Since \([(2^{4}{:}(S_{3} \times S_{3})):H_{4}] = 3,\) then \(H_{4}\) must be maximal in \(2^{4}{:}(S_{3} \times S_{3})\) (as the index is a prime number). Now the group \(2^{4}{:}(S_{3} \times S_{3})\) has 6 conjugacy classes of maximal subgroups with representatives having orders 288 (three times), 192 (twice) and 36 with respective indices 2 (three times), 3 (twice) and 16. The two maximal subgroups of order 192 are of the form \(2^{3}{:}S_{4}\) and \(2^{4}{:}D_{12}.\) The group \(2^{3}{:}S_{4}\) has 14 ordinary irreducible characters, while the group \(2^{4}{:}D_{12}\) has 13 ordinary irreducible characters. .Therefore in total for \(H_{4},\) we obtain that:

$$\begin{aligned} (H_{4}, |\mathrm {Irr}(H_{4})|) \in \{ (2^{3}{:}S_{4}, 14), (2^{4}{:}D_{12}, 13) \}. \end{aligned}$$
(2)

In Tables  4 and 5 we supply the full character tables of the two groups \(2^{3}{:}S_{4}\) and \(2^{4}{:}D_{12}\) together with the fusions of their conjugacy classes into the classes of \(A_{8}.\)

Table 4 The character table of \( 2^{3}{:}S_{4}\)
Full size table
Table 5 The character table of \(2^{4}{:}D_{12}\)
Full size table

Next we turn to look at the group \(H_{5},\) which has order 120 and index 105 in \(A_{8}.\) Again from the maximal subgroups of \(A_{8},\) it follows that \(H_{5}\) is either:

  • an index 21 subgroup of \(A_{7},\)

  • an index 6 subgroup of \(S_{6}\) or

  • an index 3 subgroup of \((A_{5} \times 3){:}2 \cong 3{:}S_{5}.\)

Now if \([A_{7} : H_{5}] = 21,\) then it follows that \(H_{5}\) is either an index 3 subgroup of \(A_{6}\) or it must be isomorphic to \(S_{5}\) (see page 10 of the Atlas). However by looking at the maximal subgroups of \(A_{6}\) (available in the Atlas), we can see that there is no subgroup of order 120 in \(A_{6}.\) Thus if \([A_{7} : H_{5}] = 21,\) then \(H_{5}\) must necessarily be \(S_{5}.\) Consider the case that \([S_{6} : H_{5}] = 6.\) The group \(S_{6}\) has 6 maximal subgroups of orders 360, 120 (twice), 72 (twice) and 72 (these maximal subgroups can be obtained easily by GAP or Magma). The subgroup of order 360 is isomorphic to \(A_{6}\), while the two groups of orders 120 are both isomorphic to \(S_{5}\) (two non-conjugate copies). Now if \([S_{6} : H_{5}] = 6,\) then \(H_{5}\) is either an index 3 subgroup of \(A_{6}\) or it must be isomorphic to \(S_{5}.\) However we mentioned that \(A_{6}\) does not contain a subgroup of order 120. Therefore \(H_{5} = S_{5}.\) Now we consider the last possibility that \(H_{5}\) an index 3 subgroup of \((A_{5} \times 3){:}2 \cong 3{:}S_{5}.\) A subgroup of \(3{:}S_{5}\) of index 3 is clearly \(S_{5}.\) Therefore in either case we obtain that \(H_{5} = S_{5}.\) In Table 6, we supply the full character table of the inertia factor group \(H_{5}\) together with the fusions of its conjugacy classes into the classes of \(A_{8}.\)

Table 6 The character table of \(H_{5} = S_{5}\)
Full size table

Next we turn to look at the group \(H_{6},\) which has order 72 and index 280 in \(A_{8}.\) Again from the maximal subgroups of \(A_{8},\) it follows that \(H_{6}\) is either:

  • an index 35 subgroup of \(A_{7},\)

  • an index 10 subgroup of \(S_{6},\)

  • an index 8 subgroup of \(2^{4}{:}(S_{3} \times S_{3})\) or

  • an index 5 subgroup of \((A_{5} \times 3){:}2 \cong 3{:}S_{5}.\)

Now if \([A_{7} : H_{6}] = 35,\) then it follows that \(H_{6}\) is either an index 5 subgroup of \(A_{6}\) or it must be isomorphic to \((A_{4} \times 3){:}2\) (see page 10 of the Atlas). However as we mentioned before that the group \(A_{6}\) does not contain a subgroup of index 5 (see page 4 of the Atlas) and therefore \(H_{6}\) must be \((A_{4} \times 3){:}2.\) Now this group has 9 ordinary irreducible characters. The second possibility is that \([S_{6} : H_{6}] = 10.\) If this is the case, then either \([A_{6} : H_{6}] = 5,\) which is not possible for the reason we mentioned before, or \(H_{6}\) must be isomorphic to \((S_{3}\times S_{3}){:}2.\) The latter group has 9 ordinary irreducible characters. The third possibility for \(H_{6}\) is that \([(2^{4}{:}(S_{3} \times S_{3})): H_{6}] = 8.\) The group \(2^{4}{:}(S_{3} \times S_{3})\) has 6 conjugacy classes of maximal subgroups, where representatives of these classes have orders 288 (three times), 192 (twice) and 36. The three maximal subgroups of order 288 are of the form \((A_{4} \times A_{4}){:}2\) (three isomorphic non-conjugate copies). Thus if \([(2^{4}{:}(S_{3} \times S_{3})): H_{6}] = 8,\) then \(H_{6}\) must be an index 4 subgroup of \((A_{4} \times A_{4}){:}2\) and consequently an index 2 subgroup of \(A_{4} \times A_{4}.\) However the maximal subgroups of \(A_{4} \times A_{4}\) are of orders 48 (four times) and 36 (twice). Thus no subgroup of index 2 of \(A_{4} \times A_{4}\) and therefore \(H_{6} \nleq (2^{4}{:}(S_{3} \times S_{3})).\) The last possibility for \(H_{6}\) is that \([(3{:}S_{5}): H_{6}] = 5.\) The group \(H_{6}\) being an index 5 of \(3{:}S_{5}\) means that it is a maximal subgroup in it. The group \(3{:}S_{5}\) has 5 maximal subgroups of orders 180, 120, 72, 60 and 36. The maximal subgroup of order 72 has the form \((3\times A_{4}){:}2,\) therefore if \([(3{:}S_{5}): H_{6}] = 5,\) then \(H_{6} = (3\times A_{4}){:}2.\) Note that \(|\mathrm {Irr}((3\times A_{4}){:}2)| = 9.\) Therefore in total for \(H_{6},\) we obtain that:

$$\begin{aligned} (H_{6}, |\mathrm {Irr}(H_{6})|) \in \{( (S_{3} \times S_{3}){:}2, 9), ((3\times A_{4}){:}2, 9) \}. \end{aligned}$$
(3)

In Tables 7 and 8 we supply the full character tables of the two groups \((S_{3}\times S_{3}){:}2\) and \((3\times A_{4}){:}2\) together with the fusions of their conjugacy classes into the classes of \(A_{8}.\)

Table 7 The character table of \((S_{3}\times S_{3}){:}2\)
Full size table
Table 8 The character table of the group \((3\times A_{4}){:}2\)
Full size table

Finally we turn to the last inertia factor group \(H_{7},\) which has order 48 and index 420 in \(A_{8}.\) From the maximal subgroups of \(A_{8},\) it follows that \(H_{7}\) is either:

  • an index 28 subgroup of \(2^{3}{:}GL(3,2),\)

  • an index 15 subgroup of \(S_{6}\) or

  • an index 12 subgroup of \(2^{4}{:}(S_{3} \times S_{3}).\)

We firstly consider the case that \([(2^{3}{:}GL(3,2)) : H_{7}] = 28.\) As we mentioned before that the group \(2^{3}{:}GL(3,2)\) has 5 conjugacy classes of maximal subgroups, where representatives of these classes have orders 192 (twice) and 168 (three times) with indices 7 (twice) and 8 (three times) respectively. We also mentioned that the two groups of order 192 are of the form \(2^{3}{:}S_{4}\) and \(2^{4}{:}D_{12}\) and we supplied their character tables in Tables 4 and 5 respectively. Now if \([(2^{3}{:}GL(3,2)) : H_{7}] = 28,\) then it follows that either \([(2^{3}{:}S_{4}) : H_{7}] = 4\) or \([(2^{4}{:}D_{12}) : H_{7}] = 4.\) Investigating the structures of the 5 maximal subgroups of \(2^{3}{:}S_{4},\) it follows that \(H_{7}\) is either an index 2 subgroup of \(2^{3}{:}A_{4}\) or isomorphic to \(2 \times S_{4}.\) However the group \(2^{3}{:}A_{4}\) does not contain a subgroup of order 48 and therefor \(H_{7}\) can only be \(2 \times S_{4},\) which has 10 ordinary irreducible characters. Similarly by investigating the 5 maximal subgroups of \(2^{4}{:}D_{12},\) it follows that \(H_{7}\) is either the group \(2^{4}{:}3,\) which has 8 ordinary irreducible characters; or \(H_{7}\) is isomorphic to \(2 \times S_{4}.\)

Now consider the second possibility is that \([(S_{6} : H_{7}] = 15.\) This is an easy case as there are two conjugacy classes of maximal subgroups of \(S_{6},\) where representatives of these classes are \(2\times S_{4}\) twice. Therefore if \([(S_{6} : H_{7}] = 15,\) then \(H_{7} = 2\times S_{4},\) where this group has 10 ordinary irreducible characters.

Finally we consider the case that \([(2^{4}{:}(S_{3} \times S_{3})) : H_{7}] = 12.\) This is a bit messy case. Since the maximal subgroups of \(2^{4}{:}(S_{3} \times S_{3})\) have orders 288 (three times), 192 (twice) and 36, it follows that \(H_{7}\) is either:

  • an index 6 subgroup of \((A_{4}\times A_{4}){:}2\),

  • an index 4 subgroup of \(2^{3}{:}S_{4}\) or

  • an index 4 subgroup of \(2^{4}{:}D_{12}.\)

In the first case that \([((A_{4}\times A_{4}){:}2) : H_{7}] = 6,\) then it follows that \(H_{7}\) is either:

  • an index 3 subgroup of \(A_{4}\times A_{4}\),

  • an index 2 subgroup of either \(2^{4}{:}S_{3}\) or

  • an index 2 subgroup of either \(2^{4}{:}6.\)

If \([(A_{4}\times A_{4}) : H_{7}] = 3,\) then \(H_{7}\) will be maximal subgroup since the index is 3 and it will follow that either \(H_{7}\) is either \(2^{2} \times A_{4}\) or \(H_{7}\) is \(2^{4}{:}3.\) The group \(2^{2} \times A_{4}\) has 16 ordinary irreducible characters, while the group \(2^{4}{:}3\) has 8 ordinary irreducible characters. If \([(2^{4}{:}S_{3}) : H_{7}] = 2,\) then \(H_{7}\) will be maximal subgroup since the index is 2 and it will follow that the group \(H_{7}\) is \(2^{4}{:}3,\) which has 8 ordinary irreducible characters. If \([(2^{4}{:}6) : H_{7}] = 2,\) then \(H_{7}\) will be maximal subgroup since the index is 2 and it will follow that the group \(H_{7}\) is \(2^{4}{:}3,\) which has 8 ordinary irreducible characters.

In the second case that \([(2^{3}{:}S_{4}) : H_{7}] = 4,\) then it follows that \(H_{7}\) is either:

  • an index 2 subgroup of either \(2^{4}{:}S_{3}\),

  • an index 2 subgroup of either \(2^{4}{:}6\) or

  • isomorphic to the group \(2 \times S_{4}.\)

In the first two cases above we obtain that \(H_{7}\) is \(2^{4}{:}3,\) where this group has 8 ordinary irreducible characters, while in the third case we can take \(H_{7}\) to be \(2 \times S_{4}\), which has 10 ordinary irreducible characters.

In the third case that \([(2^{4}{:}D_{12}) : H_{7}] = 4,\) then it follows that \(H_{7}\) is either:

  • an index 2 subgroup of either \(2^{4}{:}S_{3}\) or

  • an index 2 subgroup of either \(2^{4}{:}6.\)

In both cases above we obtain that \(H_{7}\) is \(2^{4}{:}3,\) as before, where this group has 8 ordinary irreducible characters.

Now we have exhausted all the possible cases for \(H_{7}\) and we conclude that

$$\begin{aligned} (H_{7}, |\mathrm {Irr}(H_{7})|) \in \{ (2\times S_{4}, 10), (2^{4}{:}3 ,8), (2^{2}\times A_{4}, 16) \}. \end{aligned}$$
(4)

We are now in good position to determine the structures of the remaining inertia factor groups. We start by determining \(H_{4}\) and \(H_{7}\) and after that we fix the group \(H_{6}.\)

Proposition 1

The group \(H_{4} = 2^{3}{:}S_{4}\), while \(H_{7} = 2\times S_{4}.\)

Proof

Since the group \(\overline{G} = 2^{10}{:}A_{8}\) has 81 conjugacy classes by Table 1, it follows that \(|\mathrm {Irr(\overline{G})}|=81.\) Also since the extension \(\overline{G}\) splits over N and N is an elementary abelian group, it follows that all the character tables of \(H_{1}, H_{2}, \ldots , H_{7}\) that we will use to construct the character table of \(\overline{G}\) are the ordinary ones. We also know that \(\displaystyle \sum _{i=1}^{7}|\mathrm {Irr}(H_{i})| = |\mathrm {Irr}(\overline{G})| = 81.\) Now we have \(|\mathrm {Irr}(H_{1})| = 14,\) \(|\mathrm {Irr}(H_{2})| = 11,\) \(|\mathrm {Irr}(H_{3})| = 16\) and \(|\mathrm {Irr}(H_{5})| = 7.\) Therefore it follows that

$$\begin{aligned} |\mathrm {Irr}(H_{4})| + |\mathrm {Irr}(H_{6})| + |\mathrm {Irr}(H_{7})| = 81-38 = 33. \end{aligned}$$
(5)

Now substituting the different possible pairs that are listed in Eqs. 2, 3 and 4 in Eq. 5, we can immediately deduce that \(H_{4} = 2^{3}{:}S_{4}\) and \(H_{7} = 2\times S_{4},\) while \((H_{6}, |\mathrm {Irr}(H_{6})|) \in \{( (S_{3} \times S_{3}){:}2, 9), ((3\times A_{4}){:}2, 9) \}\) as claimed. \(\square \)

In Table 9, we supply the full character table of the inertia factor group \(H_{7}\) together with the fusions of its conjugacy classes into the classes of \(A_{8}.\)

Table 9 The character table of \(H_{7} = 2\times S_{4}\)
Full size table

The last step is to determine the structure of the group \(H_{6}.\) This will be an application of the interplay between the coset analysis technique and the Fischer matrices, where this connection has been mentioned in some details in [7]. In particular, the size of the Fischer matrix correspond to a conjugacy class \([g]_{G}\) is equal to c(g),  where c(g) is the number of conjugacy classes of the full extension \(\overline{G}\) that correspond to the conjugacy class \([g]_{G}\) obtained using the coset analysis technique.

Remark 1

Recall that (see [2] for example) the rows of any Fischer matrix \(\mathcal {F}_{i}\) (corresponds to the class \([g_{i}]_{G}\)) are partitioned into submatrices correspond to the inertia factors, where there is possible fusions from the conjugacy classes of these inertia factors into the class \([g_{i}]_{G}.\)

Proposition 2

The group \(H_{6}\) is \((S_{3} \times S_{3}){:}2.\)

Proof

By Eq. 3, we have that \(H_{6} \in \{(S_{3} \times S_{3}){:}2, (3\times A_{4}){:}2 \}.\) Let us assume that \(H_{6} = (3\times A_{4}){:}2.\) From Table 1 we can see that \(\overline{G} = 2^{10}{:}A_{8}\) has three conjugacy classes correspond to the class \([g_{4}]_{A_{8}} = [3A]_{A_{8}}.\) Therefore the Fischer matrix \(\mathcal {F}_{4}\) will be a \(3 \times 3\) matrix. Now for the Fischer matrix \(\mathcal {F}_{4},\) we have one row corresponds to the first inertia factor \(H_{1} = A_{8}.\) From Table 3, we can see that there is a conjugacy class of \(H_{3}\) that fuse to the class \(g_{4} = 3A.\) Also from Table 8, we can see that there are two conjugacy classes of the group \((3\times A_{4}){:}2\) that fuse to the class \(g_{4} = 3A.\) Therefore in total the three inertia factors \(H_{1},\) \(H_{3}\) and \(H_{6}\) contribute with 4 rows to the Fischer matrix \(\mathcal {F}_{3},\) meaning that \(\mathcal {F}_{4}\) is a \(4 \times k\) matrix (for some positive integer k). But this contradicts the fact that \(\mathcal {F}_{3}\) is a \(3 \times 3\) matrix. Therefore \(H_{6} \ne (3\times A_{4}){:}2.\) Hence we deduce that \(H_{6} = (S_{3} \times S_{3}){:}2.\) \(\square \)

We conclude this section by mentioning that the groups \(H_{4},\) \(H_{6}\) and \(H_{7}\) are not maximal subgroups of \(A_{8},\) but they are maximal of some maximal subgroups of \(A_{8}\). In all the previous character tables of the inertia factor groups, we determined the fusions of the conjugacy classes of \(H_{2}, H_{3}, \ldots , H_{7}\) into the classes of \(A_{8}\) using the permutation characters of \(A_{8}\) on \(H_{2},\) \(H_{3}\) and \(H_{5}\) respectively; and the permutation characters of \(2^{3}{:}GL(3,2),\) \(S_{5}\) and \(S_{5}\) on \(H_{4},\) \(H_{6}\) and \(H_{7}\) respectively; together with the size of centralizers. We found the following proposition to be very useful in the process of determining the fusions.

Proposition 3

Let \(K_{1}\le K_{2}\le K_{3}\) and let \(\psi \) be a class function on \(K_{1}.\) Then \((\psi {\uparrow }_{K_{1}}^{K_{2}}){\uparrow }_{K_{2}}^{K_{3}} = \psi {\uparrow }_{K_{1}}^{K_{3}}.\) More generally if \(K_{1}\le K_{2}\le \cdots \le K_{n}\) is a nested sequence of subgroups of \(K_{n}\) and \(\psi \) is a class function on \(K_{1},\) then \((\psi {\uparrow }_{K_{1}}^{K_{2}}){\uparrow }_{K_{2}}^{K_{3}}\cdots {\uparrow }_{K_{n-1}}^{K_{n}} = \psi {\uparrow }_{K_{1}}^{K_{n}}.\)

Proof

See Proposition 3.5.6 of Basheer [2]. \(\square \)

4 Fischer matrices of \(\overline{G} =2^{10}{:}A_{8}\)

In this section we calculate the Fischer matrices of \(\overline{G} = 2^{10}{:}A_{8}.\) From Sect. 3 of Basheer and Moori [3] we recall that we label the top and bottom of the columns of the Fischer matrix \(\mathcal {F}_{i},\) corresponding to \(g_{i},\) by the sizes of the centralizers of \(g_{ij},\ 1\le j \le c(g_{i}),\) in \(\overline{G}\) and \(m_{ij}\) respectively. Also the rows of \(\mathcal {F}_{i}\) are partitioned into parts \(\mathcal {F}_{ik},\ 1\le k \le t,\) corresponding to the inertia factors \(H_{1}, H_{2}, \ldots , H_{t},\) where each \(\mathcal {F}_{ik}\) consists of \(c(g_{ik})\) rows correspond to the \(\alpha _{k}^{-1}-\)regular classes (those are the \(H_{k}-\)classes that fuse to class \([g_{i}]_{G}\)). Thus every row of \(\mathcal {F}_{i}\) is labeled by the pair (k, m),  where \(1\le k \le t\) and \(1\le m \le c(g_{ik}).\) In Table 1 we supplied \(|C_{\overline{G}}(g_{ij})|\) and \(m_{ij},\ 1\le i \le 14,\ 1\le j \le c(g_{i}).\) Also the fusions of classes of \(H_{2},\ H_{3}, \ldots , H_{7}\) into classes of \(G= A_{8}\) are given in Tables 2,  3, 4,  6, 7 and 9. Since the size of the Fischer matrix \(\mathcal {F}_{i}\) is \(c(g_{i}),\) it follows from Table 1 that the sizes of the Fischer matrices of \(\overline{G} = 2^{10}{:}A_{8}\) range between 1 and 16 for every \(i\in \{1,2, \ldots , 14\}.\)

We have used the arithmetical properties of the Fischer matrices, given in Proposition 3.6 of [3], to calculate some of the entries of these matrices and to build a system of algebraic equations. With the help of the symbolic mathematical package Maxima [20], we were able to solve the systems of equations and hence we have computed all the Fischer matrices of \(\overline{G},\) which we list below.

\(\mathcal {F}_{1}\)

\(g_{1}\)

 

\(g_{11}\)

\(g_{12}\)

\(g_{13}\)

\(g_{14}\)

\(g_{15}\)

\(g_{16}\)

\(g_{17}\)

\(o(g_{1j})\)

 

1

2

2

2

2

2

2

\(|C_{\overline{G}}(g_{1j})|\)

 

20643840

1376256

737280

589824

196608

49152

49152

(k, m)

\(|C_{H_{k}}(g_{1\,\,km})|\)

       

(1, 1)

20160

1

1

1

1

1

1

1

(2, 1)

1344

15

\(-1\)

15

15

\(-1\)

\(-1\)

\(-1\)

(3, 1)

576

35

−21

−5

3

11

−5

3

(4, 1)

192

105

49

−15

9

17

1

−7

(5, 1)

120

168

−56

24

−24

8

8

−8

(6, 1)

72

280

56

40

−40

−8

−8

8

(7, 1)

48

420

−28

−60

36

−28

4

4

\(m_{1j}\)

 

1

15

28

35

105

420

420

\(\mathcal {F}_{2}\)

\(g_{2}\)

 

\(g_{21}\)

\(g_{22}\)

\(g_{23}\)

\(g_{24}\)

\(g_{25}\)

\(g_{26}\)

\(g_{27}\)

\(g_{28}\)

\(g_{29}\)

\(g_{2,10}\)

\(g_{2,11}\)

\(g_{2,12}\)

\(g_{2,13}\)

\(g_{2,14}\)

\(g_{2,15}\)

\(g_{2,16}\)

\(o(g_{2j})\)

 

2

2

2

2

2

2

4

4

4

4

4

4

4

4

4

4

\(|C_{\overline{G}}(g_{2j})|\)

 

24576

24576

24576

24576

4096

4096

6144

6144

6144

6144

4096

4096

2048

1024

1024

1024

(k, m)

\(|C_{H_{k}}(g_{2\,\,km})|\)

                

(1, 1)

192

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

(2, 1)

192

1

1

1

1

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

1

1

1

\(-1\)

\(-1\)

1

(2, 2)

32

6

6

6

6

\(-2\)

\(-2\)

0

0

0

0

\(-2\)

\(-2\)

6

0

0

\(-2\)

(3, 1)

64

3

3

3

3

\(-1\)

\(-1\)

\(-3\)

\(-3\)

\(-3\)

\(-3\)

3

3

\(-1\)

1

1

\(-1\)

(3, 2)

48

4

\(-4\)

4

\(-4\)

\(-4\)

4

\(-2\)

2

\(-2\)

2

0

0

0

2

\(-2\)

0

(3, 3)

48

4

4

\(-4\)

\(-4\)

0

0

\(-4\)

\(-4\)

4

4

4

\(-4\)

0

0

0

0

(4, 1)

64

3

3

3

3

\(-1\)

\(-1\)

3

3

3

3

3

3

\(-1\)

\(-1\)

1

\(-1\)

(4, 2)

48

4

4

\(-4\)

\(-4\)

0

0

4

4

\(-4\)

\(-4\)

4

\(-4\)

0

0

0

0

(4, 3)

32

6

6

6

6

6

6

0

0

0

0

\(-2\)

\(-2\)

\(-2\)

0

0

\(-2\)

(4, 4)

16

12

\(-12\)

12

\(-12\)

4

\(-4\)

6

\(-6\)

6

\(-6\)

0

0

0

2

\(-2\)

0

(5, 1)

12

16

\(-16\)

\(-16\)

16

0

0

\(-8\)

8

8

\(-8\)

0

0

0

0

0

0

(6, 1)

12

16

\(-16\)

\(-16\)

16

0

0

8

\(-8\)

\(-8\)

8

0

0

0

0

0

0

(7, 1)

48

4

\(-4\)

4

\(-4\)

4

4

\(-4\)

4

\(-4\)

4

0

0

0

\(-2\)

0

0

(7, 2)

16

12

12

12

12

\(-4\)

\(-4\)

0

0

0

0

\(-4\)

\(-4\)

\(-4\)

0

0

4

(7, 3)

16

12

\(-12\)

12

\(-12\)

\(-4\)

\(-4\)

0

0

0

0

0

0

0

\(-2\)

4

0

(7, 4)

8

24

24

\(-24\)

\(-24\)

0

0

0

0

0

0

\(-8\)

8

0

0

0

0

\(m_{2j}\)

 

8

8

8

8

48

48

32

32

32

32

48

48

96

192

192

192

\(\mathcal {F}_{3}\)

\(g_{3}\)

 

\(g_{31}\)

\(g_{32}\)

\(g_{33}\)

\(g_{34}\)

\(g_{35}\)

\(g_{36}\)

\(g_{37}\)

\(g_{38}\)

\(g_{39}\)

\(g_{3,10}\)

\(o(g_{3j})\)

 

2

2

4

4

4

4

4

4

4

4

\(|C_{\overline{G}}(g_{3j})|\)

 

6144

2048

6144

2048

1024

1024

768

512

512

512

(k, m)

\(|C_{H_{k}}(g_{3\,\,km})|\)

          

(1, 1)

96

1

1

1

1

1

1

1

1

1

1

(2, 1)

32

3

3

3

3

\(-1\)

\(-1\)

3

\(-1\)

\(-1\)

\(-1\)

(3, 1)

96

1

1

1

1

−1

−1

−1

1

\(-1\)

1

(3, 2)

16

6

−2

\(-6\)

2

\(-4\)

4

0

\(-2\)

0

2

(4, 1)

32

3

3

3

3

1

1

\(-3\)

\(-1\)

1

\(-1\)

(4, 2)

16

6

\(-2\)

−6

2

4

\(-4\)

0

\(-2\)

0

2

(5, 1)

8

12

\(-4\)

12

−4

\(-4\)

\(-4\)

0

0

4

0

(6, 1)

12

8

8

\(-8\)

\(-8\)

0

0

0

0

0

0

(6, 2)

8

12

\(-4\)

12

\(-4\)

4

4

0

0

\(-4\)

0

(7, 1)

8

12

\(-4\)

\(-12\)

4

0

0

0

4

0

\(-4\)

\(m_{3j}\)

 

16

48

16

48

96

96

128

192

192

192

\(\mathcal {F}_{4}\)

\(g_{4}\)

 

\(g_{41}\)

\(g_{42}\)

\(g_{43}\)

\(o(g_{4j})\)

 

3

6

6

\(|C_{\overline{G}}(g_{4j})|\)

 

2880

576

288

(k, m)

\(|C_{H_{k}}(g_{4\,\,km})|\)

   

(1, 1)

180

1

1

1

(3, 1)

36

5

\(-3\)

1

(6, 1)

18

10

2

\(-2\)

\(m_{4j}\)

 

64

320

640

\(\mathcal {F}_{5}\)

\(g_{5}\)

 

\(g_{51}\)

\(g_{52}\)

\(g_{53}\)

\(g_{54}\)

\(g_{55}\)

\(g_{56}\)

\(g_{57}\)

\(g_{58}\)

\(o(g_{5j})\)

 

3

6

6

6

6

6

6

6

\(|C_{\overline{G}}(g_{5j})|\)

 

288

288

288

288

96

96

96

96

(k, m)

\(|C_{H_{k}}(g_{5\,\,km})|\)

        

(1, 1)

18

1

1

1

1

1

1

1

1

(2, 1)

6

3

3

3

3

\(-1\)

\(-1\)

\(-1\)

\(-1\)

(3, 1)

18

1

−1

−1

1

1

1

−1

\(-1\)

(3, 2)

18

1

−1

1

\(-1\)

1

−1

1

\(-1\)

(4, 1)

6

3

\(-3\)

−3

3

\(-1\)

\(-1\)

1

1

(5, 1)

6

3

3

−3

\(-3\)

\(-1\)

1

1

\(-1\)

(6, 1)

18

1

1

\(-1\)

\(-1\)

1

\(-1\)

\(-1\)

1

(7, 1)

6

3

\(-3\)

3

\(-3\)

\(-1\)

1

\(-1\)

1

\(m_{5j}\)

 

64

64

64

64

192

192

192

192

\(\mathcal {F}_{6}\)

\(g_{6}\)

 

\(g_{61}\)

\(g_{62}\)

\(g_{63}\)

\(g_{64}\)

\(g_{65}\)

\(g_{66}\)

\(g_{67}\)

\(g_{68}\)

\(g_{69}\)

\(g_{6,10}\)

\(g_{6,11}\)

\(g_{6,12}\)

\(o(g_{6j})\)

 

4

4

4

4

4

4

4

4

8

8

8

8

\(|C_{\overline{G}}(g_{6j})|\)

 

256

256

256

256

256

256

256

256

128

128

128

128

(k, m)

\(|C_{H_{k}}(g_{6\,\,km})|\)

            

(1, 1)

16

1

1

1

1

1

1

1

1

1

1

1

1

(2, 1)

16

1

1

1

1

1

1

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

(2, 2)

8

2

2

2

2

\(-2\)

\(-2\)

\(-2\)

\(-2\)

0

0

0

0

(3, 1)

16

1

1

\(-1\)

\(-1\)

1

1

−1

−1

1

1

\(-1\)

\(-1\)

(3, 2)

16

1

\(-1\)

\(-1\)

1

1

\(-1\)

1

−1

\(-1\)

1

1

\(-1\)

(3, 3)

16

1

\(-1\)

1

\(-1\)

1

\(-1\)

−1

1

1

\(-1\)

1

\(-1\)

(4, 1)

16

1

\(-1\)

\(-1\)

1

1

\(-1\)

−1

1

1

\(-1\)

\(-1\)

1

(4, 2)

16

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

1

−1

\(-1\)

1

\(-1\)

1

(4, 3)

16

1

\(-1\)

1

\(-1\)

\(-1\)

1

\(-1\)

−1

\(-1\)

\(-1\)

1

1

(4, 4)

8

2

2

\(-2\)

\(-2\)

2

\(-2\)

2

2

0

0

0

0

(7, 1)

8

2

\(-2\)

2

\(-2\)

\(-2\)

2

\(-2\)

2

0

0

0

0

(7, 2)

8

2

\(-2\)

\(-2\)

2

\(-2\)

2

2

\(-2\)

0

0

0

0

\(m_{6j}\)

 

64

64

64

64

64

64

64

64

128

128

128

128

\(\mathcal {F}_{7}\)

\(g_{7}\)

 

\(g_{71}\)

\(g_{72}\)

\(g_{73}\)

\(g_{74}\)

\(g_{75}\)

\(g_{76}\)

\(o(g_{7j})\)

 

4

4

8

8

8

8

\(|C_{\overline{G}}(g_{7j})|\)

 

64

64

64

64

32

32

(k, m)

\(|C_{H_{k}}(g_{7\,\,km})|\)

      

(1, 1)

8

1

1

1

1

1

1

(2, 1)

8

1

1

1

1

\(-1\)

\(-1\)

(3, 1)

8

1

1

−1

\(-1\)

1

\(-1\)

(4, 1)

8

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

(5, 1)

4

2

\(-2\)

2

\(-2\)

0

0

(6, 1)

4

2

\(-2\)

\(-2\)

2

0

0

\(m_{7j}\)

 

128

128

128

128

256

256

\(\mathcal {F}_{8}\)

\(g_{8}\)

 

\(g_{81}\)

\(g_{82}\)

\(o(g_{8j})\)

 

5

10

\(|C_{\overline{G}}(g_{8j})|\)

 

60

20

(k, m)

\(|C_{H_{k}}(g_{8\,\,km})|\)

  

(1, 1)

15

1

1

(5, 1)

5

3

\(-1\)

\(m_{8j}\)

 

256

768

\(\mathcal {F}_{9}\)

\(g_{9}\)

 

\(g_{91}\)

\(g_{92}\)

\(g_{93}\)

\(o(g_{9j})\)

 

6

12

12

\(|C_{\overline{G}}(g_{9j})|\)

 

48

48

24

(k, m)

\(|C_{H_{k}}(g_{9\,\,km})|\)

   

(1, 1)

12

1

1

1

(3, 1)

12

1

1

\(-1\)

(6, 1)

6

2

\(-2\)

0

\(m_{9j}\)

 

256

256

512

\(\mathcal {F}_{10}\)

\(g_{10}\)

 

\(g_{10,1}\)

\(g_{10,2}\)

\(g_{10,3}\)

\(g_{10,4}\)

\(g_{10,5}\)

\(g_{10,6}\)

\(g_{10,7}\)

\(g_{10,8}\)

\(o(g_{10j})\)

 

6

6

6

6

12

12

12

12

\(|C_{\overline{G}}(g_{10j})|\)

 

48

48

48

48

48

48

48

48

(k, m)

\(|C_{H_{k}}(g_{10\,\,km})|\)

        

(1, 1)

6

1

1

1

1

1

1

1

1

(2, 1)

6

1

1

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

(3, 1)

6

1

1

−1

−1

1

1

−1

\(-1\)

(3, 2)

6

1

−1

1

−1

1

−1

1

\(-1\)

(4, 1)

6

1

1

\(-1\)

\(-1\)

\(-1\)

\(-1\)

1

\(-1\)

(5, 1)

6

1

\(-1\)

\(-1\)

1

\(-1\)

1

1

\(-1\)

(6, 1)

6

1

\(-1\)

\(-1\)

1

1

\(-1\)

\(-1\)

1

(7, 1)

6

1

\(-1\)

1

\(-1\)

\(-1\)

1

\(-1\)

1

\(m_{10j}\)

 

128

128

128

128

128

128

128

128

\(\mathcal {F}_{11}\)

\(g_{11}\)

 

\(g_{11,1}\)

\(g_{11,2}\)

\(o(g_{11j})\)

 

7

14

\(|C_{\overline{G}}(g_{11j})|\)

 

14

14

(k, m)

\(|C_{H_{k}}(g_{11\,\,km})|\)

  

(1, 1)

7

1

1

(2, 1)

7

1

\(-1\)

\(m_{11j}\)

 

512

512

\(\mathcal {F}_{12}\)

\(g_{12}\)

 

\(g_{12,1}\)

\(g_{12,2}\)

\(o(g_{12j})\)

 

7

14

\(|C_{\overline{G}}(g_{12j})|\)

 

14

14

(k, m)

\(|C_{H_{k}}(g_{12\,\,km})|\)

  

(1, 1)

7

1

1

(2, 1)

7

1

\(-1\)

\(m_{12j}\)

 

512

512

\(\mathcal {F}_{13}\)

\(g_{13}\)

 

\(g_{13,1}\)

\(o(g_{13j})\)

 

15

\(|C_{\overline{G}}(g_{13j})|\)

 

15

(k, m)

\(|C_{H_{k}}(g_{13\,\,km})|\)

 

(1, 1)

15

1

\(m_{13j}\)

 

1024

\(\mathcal {F}_{14}\)

\(g_{14}\)

 

\(g_{14,1}\)

\(o(g_{14j})\)

 

15

\(|C_{\overline{G}}(g_{14j})|\)

 

15

(k, m)

\(|C_{H_{k}}(g_{14\,\,km})|\)

 

(1, 1)

15

1

\(m_{14j}\)

 

1024

5 Character table of \(\overline{G} = 2^{10}{:}A_{8}\)

Through Sects. 2, 3 and 4, we have determined

  • the conjugacy classes of \(\overline{G} = 2^{10}{:}A_{8}\) (Table  1),

  • the inertia factors \(H_{1},\ H_{2}, \ldots , H_{7}.\)

  • the character tables of all the inertia factor groups of G (Tables 2, 3, 4, 6, 7 and 9). In these tables we also supplied the fusions of the classes of the inertia factors \(H_{2},\ H_{3},\ldots , H_{7}\) into the classes of \(G= A_{8}.\)

  • the Fischer matrices of \(\overline{G}\) (see Sect. 4).

It follows by [2, 3] that the full character table of \(\overline{G}\) can be constructed easily in the format of Clifford–Fischer theory. This table will be partitioned into 98 parts corresponding to the 14 cosets and the 7 inertia factor groups. The full character table of \(\overline{G}\) is \(81 \times 81\) \(\mathbb {C}\)-valued matrix. In Table 10, we supply the character table of \(\overline{G}\) in the format of Clifford–Fischer Theory. Finally we would like to remark that the accuracy of this character table has been tested using GAP.

Table 10 The character table of \(\overline{G} = 2^{10}{:}A_{8}\)
Full size table