Abstract
The Suzuki simple group Sz(8) has an automorphism group 3. Using the electronic Atlas [22], the group Sz(8) : 3 has an absolutely irreducible module of dimension 12 over \({\mathbb {F}}_{2}.\) Therefore a split extension group of the form \(2^{12}{:}(Sz(8){:}3):= {\overline{G}}\) exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of \({\overline{G}}\) by analysing the maximal subgroups of Sz(8) : 3 and maximal of the maximal subgroups of Sz(8) : 3 together with various other information. It turns out that the character table of \({\overline{G}}\) is a \(43 \times 43\) complex valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 7.
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1 Introduction
Let n be a non-negative integer and \(q= 2^{1+2n}\) and \(r= 2^{n}.\) In 1960, M. Suzuki constructed a new type of groups as subgroups of SL(4, q) generated by certain explicit matrices. These infinite family of groups nowadays are known as Suzuki groups and denoted by Sz(q). For any \(q= 2^{1+2n},\) \(n\in {\mathbb {N}},\) the Suzuki group Sz(q) is simple and has order \(q^{2}(q^{2}+1)(q-1).\) The order of Sz(q) is always divisible by 5, but not 3. A remarkable point here is that the Suzuki groups are the only non-abelian simple groups whose order is not divisible by 3. The Schur multiplier is trivial for all \(n > 1,\) and for \(n=1,\) i.e., for the group Sz(8), the Schur multiplier is isomorphic to the Klein 4-group \({\mathbb {V}}_{4} \cong 2^{2}.\) The outer automorphism group of Sz(q) is cyclic of order \(2n+1.\) In the special case, the group Sz(8) has order \(29120 = 2^{6} \times 5 \times 7 \times 13\) and has outer automorphism group isomorphic to the cyclic group \({\mathbb {Z}}_{3}.\) Therefore the group Sz(8) : 3 exists and using the electronic Atlas [22], we can see that both Sz(8) and Sz(8) : 3 can be represented in terms of matrices with small dimensions over finite fields, or in terms of permutations on 65 points. More precisely one can see that the group Sz(8) : 3 has a 12-dimensional absolutely irreducible module over \({\mathbb {F}}_{2}.\) Therefore a split extension group of the form \(2^{12}{:}(Sz(8){:}3):= {\overline{G}}\) exists. 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 Sz(8) : 3, 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 (see [1, 19]). The most interesting part is the determination of the inertia factor groups, where there are three inertia factor groups, namely \(H_{1} = Sz(8){:}3\), \(H_{2}\) and \(H_{3}.\) The main method used to determine the structures of \(H_{2}\) and \(H_{3},\) is by analysing the maximal subgroups of Sz(8) : 3 and maximal of these maximal subgroups. Sometimes we consider the third level of maximal subgroups of Sz(8) : 3. We ended up with finding that \(H_{2}\) and \(H_{3}\) are groups of the forms \(2^{3+3}{:}3\) and \(2\times A_{4}\) respectively. The Fischer matrices of \({\overline{G}}\) have all been determined in this paper and their sizes range between 1 and 7. The character table of \({\overline{G}}\) is a \(43 \times 43\) complex valued matrix and it is partitioned into 51 parts corresponding to the 3 inertia factor groups and the 17 conjugacy classes of \(G = Sz(8){:}3.\) The character table of any finite group extension \({\overline{G}} = N{\cdot }G\) (here N is the kernel of the extension and G is isomorphic to \({\overline{G}}/N\)) produced by Clifford-Fischer Theory is in a special format that could not be achieved by direct computations using GAP [17] or Magma [13]. Also 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 [22], we can see that Sz(8) : 3 has an absolutely irreducible module of dimension 12 over \({\mathbb {F}}_{2}.\) The following two elements \(g_{1}\) and \(g_{2}\) are \(12 \times 12\) matrices over \({\mathbb {F}}_{2}\) that generate Sz(8) : 3.
where \(o(g_{1}) = 2\) and \(o(g_{2}) = 3.\)
Using the above two generators of Sz(8) : 3 together with few GAP commands we were able to construct our split extension group \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\) in terms of \(13 \times 13\) matrices over \({\mathbb {F}}_{2}.\) The following three elements \({\overline{g}}_{1},\) \({\overline{g}}_{2}\) and \({\overline{g}}_{3}\) generate \({\overline{G}}\).
where \(o({\overline{g}}_{1}) =3,\) \(o({\overline{g}}_{2}) = 2\) and \(o({\overline{g}}_{3}) = 2.\)
For the sake of computations, we used few GAP commands to convert the representation of our group \({\overline{G}}\) from matrix into permutation representation, where we were able to represent \({\overline{G}}\) in terms of the set \(\{1,2, \ldots ,4096\}.\)
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 4096 and 119275520. The normal subgroup of order 4096 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 Sz(8) : 3 and any of these two complements together with N gives the split extension in consideration.
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, 14, 16].
2 Conjugacy classes of \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\)
In this section we compute the conjugacy classes of the group \({\overline{G}}\) using the coset analysis technique (see Basheer [1], Basheer and Moori [2, 3, 5] or Moori [19] and [20] for more details) as we are interested to organize the classes of \({\overline{G}}\) corresponding to the classes of Sz(8) : 3. Firstly note that Sz(8) : 3 has 17 conjugacy classes (see the Atlas [22] or Table 4 of this paper). Corresponding to these 17 classes of Sz(8) : 3, we obtained 43 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 Sz(8) : 3 of N in \({\overline{G}}\). In particular, the action of N on the identity coset N produces 4096 orbits each consists of singleton. Thus for \({\overline{G}},\) we have \(k_{1}= 4096.\)
-
\(f_{ij}\) 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 \(C_{G}(1_{G})\) \(= G = Sz(8){:}3\) on the orbits \(Q_{1}, Q_{2},\ldots ,Q_{k}\) affords three orbits of lengths 1, 455 and 3640 (with corresponding point stabilizers Sz(8) : 3, \(2^{3+3}{:}3\) and \(2\times A_{4}\). Thus \(f_{11} = 1,\ f_{12} = 455\) and \(f_{13} = 3640.\)
-
\(m_{ij}\)’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.
3 Inertia factor groups of \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\)
We have seen in Sect. 2 that the action of \({\overline{G}}\) on N produced three orbits of lengths 1, 455 and 3640. By a theorem of Brauer (for example see Theorem 5.1.1 of Basheer [1]), it follows that the action of \({\overline{G}}\) on \(\textrm{Irr}(N)\) will also produce three orbits of lengths 1, r and s, where \(1+r+s = |\textrm{Irr}(N)| = 4096;\) that is
The values of r and s will be determined through deep investigation on the maximal subgroups of Sz(8) : 3 or maximal of the maximal subgroups of Sz(8) : 3 together with various information including the sizes of the Fischer matrices, fusions of the the conjugacy classes of some subgroups into the group Sz(8) : 3 and other information. In Table 2 we supply the maximal subgroups of Sz(8) : 3, where we need these subgroups in the process of the determination of \(H_{2}\) and \(H_{3}.\)
Firstly since 1, r and s are the lengths of the orbits on the action of \({\overline{G}}\) on N (which can be reduced to the action of G on N), it follows that \([G:H_{1}] = 1,\) \([G:H_{2}] = r\) and \([G:H_{3}] = s,\) where \(H_{1},\ H_{2}\) and \(H_{3}\) are the inertia factor groups in \(G = Sz(8){:}3.\) It follows that \(H_{1} = G = Sz(8){:}3\) and \(r,s\mid |G|;\) that is r, s|87360. Now 87360 has 112 positive divisors, where 96 divisors are less than 4095. Out of these 96 divisors, only two pairs (r, s) satisfy Eq. (2). These are the pairs:
Here we do not distinguish between the pair (r, s) and (s, r) and therefore we excluded the other two pairs (3640, 455) and (2730, 1365) from our consideration and we restrict ourselves only to those in Eq. (3). Another point that we put in mind is that since the extension \({\overline{G}}\) splits over N and N is an elementary abelian group, it follows that all the character tables of \(H_{1},\ H_{2}\) and \(H_{3}\) that we will use to construct the character table of \({\overline{G}}\) are the ordinary ones. From Tables 1 and 4 we have \(|\textrm{Irr}({\overline{G}})| = 43\) and \(|\textrm{Irr}(H_{1})| = |\textrm{Irr}(G)| = |\textrm{Irr}(Sz(8){:}3)|= 17.\) Since \(\sum _{i=1}^{3}|\textrm{Irr}(H_{i})| = |\textrm{Irr}({\overline{G}})| = 43,\) we have \(|\textrm{Irr}(H_{1})| + |\textrm{Irr}(H_{2})| + |\textrm{Irr}(H_{3})| = |\textrm{Irr}({\overline{G}})|= 43,\) that is
Our next task is to show that \((r,s) = (455, 3640)\) and that the action of \({\overline{G}}\) on \(\textrm{Irr}(N)\) will be dual to the action of \({\overline{G}}\) on the classes of N. This will be achieved by excluding the other possible pair by getting a contradiction to some fact, which we show in the next proposition.
Proposition 1
\((r,s) \ne (1365, 2730).\)
Proof
For the purpose of contradiction assume \((r,s) = (1365, 2730);\) that is \(r = 1365\) and \(s = 2730\) (or \([Sz(8){:}3:H_{2}] = 1365\) and \([Sz(8){:}3:H_{3}] = 2730\)) and consequently \(|H_{2}| = 64\) and \(|H_{3}| = 32.\) By looking at the maximal subgroups of Sz(8) : 3, given in Table 2, it follows that \(H_{2}\) is either an index 455 subgroup of Sz(8) or an index 21 subgroup of \(2^{3+3}{:}(7{:}3).\) If \(H_{2}\) is an index 455 subgroup of Sz(8), then by looking at the maximal subgroups Sz(8), available in the Atlas, it follows that \(H_{2}\) must be an index 7 subgroup of \(2^{3+3}{:}7.\) In this case we can see clearly that \(H_{2}\) will be isomorphic to the group \(2^{3+3}.\) On the other hand, if \(H_{2}\) is an index 21 subgroup of \(2^{3+3}{:}(7{:}3),\) then it will also be isomorphic to the group \(2^{3+3}.\) In Table 3 we list the character table of the group \(2^{3+3}\) together with the fusions of the conjugacy classes of this group into the classes of Sz(8) : 3. The interplay between the coset analysis and Clifford-Fischer Theory was mentioned in some details in [6]. 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. Now from Table 1 we can see that \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\) has four conjugacy classes correspond to the class \([g_{5}]_{Sz(8){:}3} = [4A]_{Sz(8){:}3}.\) Therefore the Fischer matrix \({\mathcal {F}}_{5}\) will be a \(4 \times 4\) matrix. We also know that 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 \([g_{i}]_{G}.\) For the Fischer matrix \({\mathcal {F}}_{5},\) which we found to be of size 4, we have one row corresponds to the first inertia factor \(H_{1} = Sz(8){:}3.\) From Table 3, we can see that there are 7 conjugacy classes of \(H_{2}\) that fuse to the class \(g_{5} = 4A.\) Thus the contribution of \(H_{1}\) and \(H_{2}\) together to \({\mathcal {F}}_{5}\) will be 8 rows, which is a contradiction to the fact that \({\mathcal {F}}_{5}\) is a \(4 \times 4\) matrix. Therefore \(H_{2}\) can not be the group \(2^{3+3}\) and we deduce that \((r,s) = (1365, 2730)\) is not the required pair. \(\square \)
Corollary 2
The action of Sz(8) : 3 on \(\textrm{Irr}(2^{12})\) is dual to the action of Sz(8) : 3 on the conjugacy classes of \(N =2^{12}.\)
Proof
The application of Eq. (3) and Proposition 1 shows that \((r,s) = (455, 3640)\) and it follows that the action of Sz(8) : 3 on \(\textrm{Irr}(2^{12})\) is dual to the action of Sz(8) : 3 on the conjugacy classes of \(N =2^{12}\) as claimed. \(\square \)
Proposition 3
The inertia factor groups have the forms \(2^{3+3}{:}3\) and \(2\times A_{4}.\)
Proof
We found that the orbit lengths on the action of Sz(8) : 3 on \(\textrm{Irr}(2^{12})\) are 1, 455 and 3640. It follows that \([G:H_{1}] = 1,\) \([G:H_{2}] = 455\) and \([G:H_{3}] = 3640\) and consequently \(H_{1} = G = Sz(8){:}3,\) \(|H_{2}| = 192\) and \(|H_{3}| = 24.\) By Eq. (4) we also have \(|\textrm{Irr}(H_{2})| + |\textrm{Irr}(H_{3})| = 26.\) Now we investigate the maximal subgroups of Sz(8) : 3 to locate \(H_{2}\) and \(H_{3}.\) Since \(|H_{2}| = 192\) and by looking at the maximal subgroups of Sz(8) : 3, given in Table 2, it follows that \(H_{2}\) must be an index 7 subgroup of \(2^{3+3}{:}(7{:}3)\) and since the index is 7, then \(H_{2}\) must be maximal in \(2^{3+3}{:}(7{:}3).\) Now the group \(2^{3+3}{:}(7{:}3)\) has 3 maximal subgroups of orders 448, 192 and 168. The group of order 192 has the structure \(2^{3+3}{:}3\) and therefore \(H_{2}\) is isomorphic to group \(2^{3+3}{:}3,\) which has 18 irreducible characters. Using this together with Eq. (4) we deduce that the third inertia factor group that we are looking for must have 8 ordinary irreducible characters.
Next we consider \(H_{3}.\) Since \(|H_{3}| = 24\) and by looking at the maximal subgroups of Sz(8) : 3, given in Table 2, it follows that \(H_{3}\) is an index 56 subgroup of \(2^{3+3}{:}(7{:}3).\) From the above we know that \(2^{3+3}{:}(7{:}3)\) has 3 maximal subgroups of orders 448, 192 and 168 with respective structures \(2^{3+3}{:}7,\) \(2^{3+3}{:}3\) and \(2^{3}{:}(7{:}3)\). Therefore \(H_{3}\) is either:
-
an index 8 subgroup of \(2^{3+3}{:}3\) or
-
an index 7 subgroup of \(2^{3}{:}(7{:}3).\)
Now the group \(2^{3+3}{:}3\) has 3 conjugacy classes of maximal subgroups represented by \(2^{3+2}{:}3,\) \(2^{3+3}\) and \(4 \times A_{4}\) with respective orders 96, 64 and 48. Since \(24 \not \mid 64,\) then it is clear that either \(H_{3} \le 2^{3+2}{:}3\) with index 4 or \(H_{3} \le 4 \times A_{4}\) with index 2. The group \(2^{3+2}{:}3\) has two conjugacy classes of maximal subgroups represented by \((2 \times 4){:}4\) and \(2 \times A_{4}\) with orders 32 and 24. Therefore if \(H_{3} \le 2^{3+2}{:}3\) with index 4, then it must be isomorphic to the group \(2 \times A_{4},\) which has 8 irreducible characters. On the other hand in the case that \(H_{3} \le 4 \times A_{4}\) with index 2, we notice that the group \(4 \times A_{4}\) has 3 conjugacy classes of maximal subgroups represented by \(2 \times A_{4},\) \(4 \times 2 \times 2\) and \({\mathbb {Z}}_{12}\) with respective orders 24, 16 and 12. Therefore if \(H_{3} \le 4 \times A_{4}\) with index 2, then it must be isomorphic to the group \(2 \times A_{4},\) which has 8 irreducible characters. Next we turn to the case that \(H_{3}\) is an index 7 subgroup of \(2^{3}{:}(7{:}3).\) In this case it is clear that \(H_{3}\) is maximal in \(2^{3}{:}(7{:}3)\) since the index is prime. Now the group \(2^{3}{:}(7{:}3)\) has 3 conjugacy classes of maximal subgroups represented by \(2^{3}{:}7,\) \(2 \times A_{4}\) and 7 : 3 with respective orders 56, 24 and 21. Therefore if \(H_{3} \le 2^{3}{:}(7{:}3)\) with index 7, then it must be isomorphic to the group \(2 \times A_{4},\) which has 8 irreducible characters. In either case we can see that the group \(H_{3}\) has the structure \(2 \times A_{4}.\) This completes the investigation on the two inertia factor groups \(H_{2}\) and \(H_{3}.\) \(\square \)
As subgroups of the full extension \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\) that is generated by \({\overline{g}}_{1},\) \({\overline{g}}_{2}\) and \({\overline{g}}_{3}\) given in Section 1, the two inertia factor groups \(H_{2} = 2^{3+3}{:}3\) and \(H_{3} = 2\times A_{4}\) are generated as follows: \(H_{2} = \left<\alpha _{1}, \alpha _{2}, \alpha _{3}\right>\) and \(H_{3} = \left<\beta _{1}, \beta _{2}\right>,\) where
The character table of the simple Suzuki group Sz(8) is available in the Atlas and thus the extension Sz(8) : 3 can easily be constructed using Clifford-Fischer Theory. However we used the two generators \(g_{1}\) and \(g_{2}\) given in Sect. 1 together with GAP to construct the character table of Sz(8) : 3, which we list in Table 4. Also at the end of this paper we list the character tables of \(H_{2}\) and \(H_{3}\) that we can easily obtain through GAP or Magma. Recall that \(H_{2}\) and \(H_{3}\) are not maximal subgroups of Sz(8) : 3, but they are maximal of some maximal subgroups of Sz(8) : 3. We determined the fusions of the conjugacy classes of \(H_{2}\) and \(H_{3}\) into the classes Sz(8) : 3 using the permutation characters of Sz(8) : 3 on \(2^{3+3}{:}(7 {:}3)\) and \(2^{3+3}{:}3;\) the permutation characters of \(2^{3+3}{:}(7 {:}3)\) and \(2^{3+3}{:}3\) on \(H_{2}\) and \(H_{3}\) respectively; together with the size of centralizers. We found the following proposition to be very useful in the process of determining the fusions.
Proposition 4
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 [1]. \(\square \)
In Tables 5 and 6 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 Sz(8) : 3.
4 Fischer matrices of \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\)
In this section we calculate the Fischer matrices of \({\overline{G}} = 2^{12}{:}(Sz(8){:}3).\) From Sect. 3 of Basheer and Moori [2] 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 \(m_{ij},\ 1\le i \le 17,\ 1\le j \le c(g_{i}).\) Also the fusions of classes of \(H_{2}\) and \(H_{3}\) into classes of G are given in Tables 5 and 6 respectively. 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 \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\) range between 1 and 7 for every \(i\in \{1,2, \ldots , 17\}.\)
We have used the arithmetical properties of the Fischer matrices, given in Proposition 3.6 of [2], 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 [18], 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.
\({\mathcal {F}}_{1}\) | ||||
|---|---|---|---|---|
\(g_{1}\) | \(g_{11}\) | \(g_{12}\) | \(g_{13}\) | |
\(o(g_{1j})\) | 1 | 2 | 2 | |
\(|C_{{\overline{G}}}(g_{1j})|\) | 357826560 | 786432 | 98304 | |
(k, m) | \(|C_{H_{k}}(g_{1km})|\) | |||
(1, 1) | 87360 | 1 | 1 | 1 |
(2, 1) | 192 | 455 | \(-57\) | 7 |
(3, 1) | 24 | 3640 | 56 | \(-8\) |
\(m_{1j}\) | 1 | 455 | 3640 | |
\({\mathcal {F}}_{2}\) | ||||||||
|---|---|---|---|---|---|---|---|---|
\(g_{2}\) | \(g_{21}\) | \(g_{22}\) | \(g_{23}\) | \(g_{24}\) | \(g_{25}\) | \(g_{26}\) | \(g_{27}\) | |
\(o(g_{2j})\) | 2 | 4 | 4 | 4 | 4 | 4 | 4 | |
\(|C_{{\overline{G}}}(g_{2j})|\) | 12288 | 12288 | 4096 | 4096 | 1536 | 512 | 512 | |
(k, m) | \(|C_{H_{k}}(g_{2km})|\) | |||||||
(1, 1) | 192 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
(2, 1) | 192 | 1 | 1 | 1 | 1 | \(-1\) | 1 | \(-1\) |
(2, 2) | 64 | 3 | 3 | 3 | 3 | \(-3\) | \(-1\) | 1 |
(2, 3) | 64 | 3 | 3 | 3 | 3 | 3 | \(-1\) | \(-1\) |
(3, 1) | 24 | 8 | \(-8\) | 8 | \(-8\) | 0 | 0 | 0 |
(3, 2) | 8 | 24 | 24 | \(-8\) | \(-8\) | 0 | 0 | 0 |
(3, 3) | 8 | 24 | \(-24\) | \(-8\) | 8 | 0 | 0 | 0 |
\(m_{2j}\) | 64 | 64 | 192 | 192 | 512 | 1536 | 1536 | |
\({\mathcal {F}}_{3}\) | ||||
|---|---|---|---|---|
\(g_{3}\) | \(g_{31}\) | \(g_{32}\) | \(g_{33}\) | |
\(o(g_{3j})\) | 3 | 6 | 6 | |
\(|C_{{\overline{G}}}(g_{3j})|\) | 960 | 192 | 96 | |
(k, m) | \(|C_{H_{k}}(g_{3km})|\) | |||
(1, 1) | 60 | 1 | 1 | 1 |
(2, 1) | 12 | 5 | \(-3\) | 1 |
(3, 1) | 6 | 10 | 2 | \(-2\) |
\(m_{3j}\) | 256 | 1280 | 2560 | |
\({\mathcal {F}}_{4}\) | ||||
|---|---|---|---|---|
\(g_{4}\) | \(g_{41}\) | \(g_{42}\) | \(g_{43}\) | |
\(o(g_{4j})\) | 3 | 6 | 6 | |
\(|C_{{\overline{G}}}(g_{4j})|\) | 960 | 192 | 96 | |
(k, m) | \(|C_{H_{k}}(g_{4km})|\) | |||
(1, 1) | 60 | 1 | 1 | 1 |
(2, 1) | 12 | 5 | \(-3\) | 1 |
(3, 1) | 6 | 10 | 2 | \(-2\) |
\(m_{4j}\) | 256 | 1280 | 2560 | |
\({\mathcal {F}}_{5}\) | |||||
|---|---|---|---|---|---|
\(g_{5}\) | \(g_{51}\) | \(g_{52}\) | \(g_{53}\) | \(g_{54}\) | |
\(o(g_{5j})\) | 4 | 8 | 8 | 8 | |
\(|C_{{\overline{G}}}(g_{5j})|\) | 384 | 384 | 128 | 128 | |
(k, m) | \(|C_{H_{k}}(g_{5km})|\) | ||||
(1, 1) | 48 | 1 | 1 | 1 | 1 |
(2, 1) | 48 | 1 | \(-1\) | 1 | \(-1\) |
(2, 2) | 16 | 3 | \(-3\) | \(-1\) | 1 |
(2, 3) | 16 | 3 | 3 | \(-1\) | \(-1\) |
\(m_{5j}\) | 512 | 512 | 1536 | 1536 | |
\({\mathcal {F}}_{6}\) | |||||
|---|---|---|---|---|---|
\(g_{6}\) | \(g_{61}\) | \(g_{62}\) | \(g_{63}\) | \(g_{64}\) | |
\(o(g_{6j})\) | 4 | 8 | 8 | 8 | |
\(|C_{{\overline{G}}}(g_{6j})|\) | 384 | 384 | 128 | 128 | |
(k, m) | \(|C_{H_{k}}(g_{6km})|\) | ||||
(1, 1) | 48 | 1 | 1 | 1 | 1 |
(2, 1) | 48 | 1 | \(-1\) | 1 | \(-1\) |
(2, 2) | 16 | 3 | \(-3\) | \(-1\) | 1 |
(2, 3) | 16 | 3 | 3 | \(-1\) | \(-1\) |
\(m_{6j}\) | 512 | 512 | 1536 | 1536 | |
\({\mathcal {F}}_{7}\) | ||
|---|---|---|
\(g_{7}\) | \(g_{71}\) | |
\(o(g_{7j})\) | 5 | |
\(|C_{{\overline{G}}}(g_{7j})|\) | 15 | |
(k, m) | \(|C_{H_{k}}(g_{7km})|\) | |
(1, 1) | 15 | 1 |
\(m_{7j}\) | 4096 | |
\({\mathcal {F}}_{8}\) | ||||
|---|---|---|---|---|
\(g_{8}\) | \(g_{81}\) | \(g_{82}\) | \(g_{83}\) | |
\(o(g_{8j})\) | 6 | 12 | 12 | |
\(|C_{{\overline{G}}}(g_{8j})|\) | 48 | 48 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{8km})|\) | |||
(1, 1) | 12 | 1 | 1 | 1 |
(2, 1) | 12 | 1 | 1 | \(-1\) |
(3, 1) | 6 | 2 | \(-2\) | 0 |
\(m_{8j}\) | 1024 | 1024 | 2048 | |
\({\mathcal {F}}_{9}\) | ||||
|---|---|---|---|---|
\(g_{9}\) | \(g_{91}\) | \(g_{92}\) | \(g_{93}\) | |
\(o(g_{9j})\) | 6 | 12 | 12 | |
\(|C_{{\overline{G}}}(g_{9j})|\) | 48 | 48 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{9km})|\) | |||
(1, 1) | 12 | 1 | 1 | 1 |
(2, 1) | 12 | 1 | 1 | \(-1\) |
(3, 1) | 6 | 2 | \(-2\) | 0 |
\(m_{9j}\) | 1024 | 1024 | 2048 | |
\({\mathcal {F}}_{10}\) | ||
|---|---|---|
\(g_{10}\) | \(g_{10,1}\) | |
\(o(g_{10j})\) | 7 | |
\(|C_{{\overline{G}}}(g_{10j})|\) | 7 | |
(k, m) | \(|C_{H_{k}}(g_{10km})|\) | |
(1, 1) | 7 | 1 |
\(m_{10j}\) | 4096 | |
\({\mathcal {F}}_{11}\) | |||
|---|---|---|---|
\(g_{11}\) | \(g_{11,1}\) | \(g_{11,2}\) | |
\(o(g_{11j})\) | 12 | 24 | |
\(|C_{{\overline{G}}}(g_{11j})|\) | 24 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{11km})|\) | ||
(1, 1) | 24 | 1 | 1 |
(2, 1) | 24 | 1 | \(-1\) |
\(m_{11j}\) | 2048 | 2048 | |
\({\mathcal {F}}_{12}\) | |||
|---|---|---|---|
\(g_{12}\) | \(g_{12,1}\) | \(g_{12,2}\) | |
\(o(g_{12j})\) | 12 | 24 | |
\(|C_{{\overline{G}}}(g_{12j})|\) | 24 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{12km})|\) | ||
(1, 1) | 24 | 1 | 1 |
(2, 1) | 24 | 1 | \(-1\) |
\(m_{12j}\) | 2048 | 2048 | |
\({\mathcal {F}}_{13}\) | |||
|---|---|---|---|
\(g_{13}\) | \(g_{13,1}\) | \(g_{13,2}\) | |
\(o(g_{13j})\) | 12 | 24 | |
\(|C_{{\overline{G}}}(g_{13j})|\) | 24 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{13km})|\) | ||
(1, 1) | 24 | 1 | 1 |
(2, 1) | 24 | 1 | \(-1\) |
\(m_{13j}\) | 2048 | 2048 | |
\({\mathcal {F}}_{14}\) | |||
|---|---|---|---|
\(g_{14}\) | \(g_{14,1}\) | \(g_{14,2}\) | |
\(o(g_{14j})\) | 12 | 24 | |
\(|C_{{\overline{G}}}(g_{14j})|\) | 24 | 24 | |
(k, m) | \(|C_{H_{k}}(g_{14km})|\) | ||
(1, 1) | 24 | 1 | 1 |
(2, 1) | 24 | 1 | \(-1\) |
\(m_{14j}\) | 2048 | 2048 | |
\({\mathcal {F}}_{15}\) | ||
|---|---|---|
\(g_{15}\) | \(g_{15,1}\) | |
\(o(g_{15j})\) | 13 | |
\(|C_{{\overline{G}}}(g_{15j})|\) | 13 | |
(k, m) | \(|C_{H_{k}}(g_{15km})|\) | |
(1, 1) | 13 | 1 |
\(m_{15j}\) | 4096 | |
\({\mathcal {F}}_{16}\) | ||
|---|---|---|
\(g_{16}\) | \(g_{16,1}\) | |
\(o(g_{16j})\) | 15 | |
\(|C_{{\overline{G}}}(g_{16j})|\) | 15 | |
(k, m) | \(|C_{H_{k}}(g_{16km})|\) | |
(1, 1) | 15 | 1 |
\(m_{16j}\) | 4096 | |
\({\mathcal {F}}_{17}\) | ||
|---|---|---|
\(g_{17}\) | \(g_{17,1}\) | |
\(o(g_{17j})\) | 15 | |
\(|C_{{\overline{G}}}(g_{17j})|\) | 15 | |
(k, m) | \(|C_{H_{k}}(g_{17km})|\) | |
(1, 1) | 15 | 1 |
\(m_{17j}\) | 4096 | |
5 Character table of \({\overline{G}} =2^{12}{:}(Sz(8){:}3)\)
Through Sects. 2, 3 and 4, we have determined
-
the conjugacy classes of \({\overline{G}} = 2^{12}{:}(Sz(8){:}3)\) (Table 1),
-
the inertia factors \(H_{1},\ H_{2}\) and \(H_{3}.\)
-
the character tables of all the inertia factor groups of G (Tables 4, 5 and 6). In Tables 5 and 6 we also supplied the fusions of the classes of the groups \(H_{2}\) and \( H_{3}\) into classes of G.
-
the Fischer matrices of \({\overline{G}}\) (see Sect. 4).
It follows by [1, 2] 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 51 parts corresponding to the 17 cosets and the three inertia factor groups. The full character table of \({\overline{G}}\) is \(43 \times 43\) \({\mathbb {C}}\)-valued matrix. In Table 7, 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.
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Appendix
Appendix
In Table 4, \(A = -\frac{1}{2}-i\frac{\sqrt{3}}{2},\) \(B = -2-2i\sqrt{3}\) and \(C = -E(12)^{7}.\)
In Table 5, \(A = -\frac{1}{2}-i\frac{\sqrt{3}}{2}\) and \(B = E(12)^{7}.\)
In Table 6, \(A = -\frac{1}{2}-i\frac{\sqrt{3}}{2}\)
In Table 7, \(A = -\frac{1}{2}-i\frac{\sqrt{3}}{2}, \) \(B = -2+2i\sqrt{3},\) \(C= -\frac{5}{2}+5i\frac{\sqrt{3}}{2},\) \(D= \frac{3}{2}-3i\frac{\sqrt{3}}{2}, \) \(E = -5+5i\sqrt{3},\) \(F = -1+i\sqrt{3}\) and \(G = -E(12)^7\).
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Basheer, A.B.M. On a group extension involving the Suzuki group Sz(8). Afr. Mat. 34, 96 (2023). https://doi.org/10.1007/s13370-023-01130-z
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DOI: https://doi.org/10.1007/s13370-023-01130-z