# ATLAS: Suzuki group Sz(8)

Order = 29120 = 26.5.7.13.
Mult = 22.
Out = 3.

### Standard generators

Standard generators of Sz(8) are a and b where a has order 2, b has order 4, ab has order 5, abb has order 7 and ababbbabb has order 7. The condition that abb has order 7 is redundant.
Standard generators of a particular double cover 2.Sz(8) are preimages A and B where AB has order 5, ABB has order 7 and ABABBBABB has order 7. Note that if (a, b) is fixed, then these relations only hold in one of the three double covers.
Standard generators of the cover 22.Sz(8) are preimages A and B where AB has order 5 and ABB has order 7.

Standard generators of Sz(8):3 are c and d where c has order 2, d has order 3, cd has order 15, and cdcdcdcddcdcddcdd has order 6. This last condition distinguishes the pair (c, d) from the pair (c, dd), and thereby distinguishes the two classes of elements of order 3. Indeed, d cycles (7A, 7B, 7C) in that order.
Standard generators of 22.Sz(8):3 are preimages C and D where CDCDD has order 13.

### Note

The old definitions of standard generators for Sz(8) and covers (which were designed with certain implementations of the MeatAxe in mind) is given below. The definitions below give the same generators up to automorphisms as the new definitions above.

Standard generators of Sz(8) are a and b where a has order 2, b has order 4, ab has order 5, ababb has order 7 and abababbababbabb has order 13.
Standard generators of a particular double cover 2.Sz(8) are preimages A and B where AB has order 5, ABABB has order 7 and ABABABBABABBABB has order 13. Note that if (a, b) is fixed, then these relations only hold in one of the three double covers.
Standard generators of 22.Sz(8) are preimages A and B where AB has order 5, ABABB has order 7.

### Generality

For the sake of labelling of characters in accordance with the ABC, we decree that abb is in class 7A and abababbababbabb is in class 13B.

### Automorphisms

An outer automorphism of Sz(8) of order 3 can be realised by mapping (a, b) to ((ab)^-4a(ab)^4, (abb)^-4b(abb)^4).
With the ATLAS notation for conjugacy classes, this cycles the classes (7A, 7B, 7C) in that order.

### Representations

The representations of Sz(8) available are
• Essentially all faithful irreducibles in characteristic 2.
• a and b as 4 × 4 matrices over GF(8) - the natural representation.
• a and b as 16 × 16 matrices over GF(8).
• a and b as 64 × 64 matrices over GF(2) - the Steinberg representation.
• Essentially all faithful irreducibles in characteristic 5.
• a and b as 14 × 14 matrices over GF(5).
• a and b as 35 × 35 matrices over GF(5).
• a and b as 35 × 35 matrices over GF(5).
• a and b as 35 × 35 matrices over GF(5).
• a and b as 65 × 65 matrices over GF(125).
• Essentially all faithful irreducibles in characteristic 7.
• a and b as 14 × 14 matrices over GF(49).
• a and b as 64 × 64 matrices over GF(7).
• a and b as 91 × 91 matrices over GF(7).
• a and b as 105 × 105 matrices over GF(7) - becomes 35abc over GF(343).
• All faithful irreducibles in characteristic 13.
• a and b as 14 × 14 matrices over GF(13).
• a and b as 14 × 14 matrices over GF(13).
• a and b as 35 × 35 matrices over GF(13).
• a and b as 65 × 65 matrices over GF(13).
• a and b as 65 × 65 matrices over GF(13).
• a and b as 65 × 65 matrices over GF(13).
• a and b as 91 × 91 matrices over GF(13).
• All primitive permutation representations.
• a and b as permutations on 65 points.
• a and b as permutations on 560 points.
• a and b as permutations on 1456 points.
• a and b as permutations on 2080 points.
The representations of 2.Sz(8) available are as follows.
(They have now been adjusted to ensure that they are all the same double cover!)
• A and B as permutations on 1040 points.
• A and B as 128 × 128 matrices over GF(2).
• A and B as 8 × 8 matrices over GF(5).
• A and B as 40 × 40 matrices over GF(7).
• A and B as 16 × 16 matrices over GF(13).
• A and B as 24 × 24 matrices over GF(13).
The representations of Sz(8).3 available are
• All faithful irreducibles in characteristic 2 (up to tensoring with linear characters).
• c and d as 12 × 12 matrices over GF(2).
• c and d as 48 × 48 matrices over GF(2).
• c and d as 64 × 64 matrices over GF(2).
• c and d as 14 × 14 matrices over GF(5).
• c and d as 14 × 14 matrices over GF(49).
• c and d as 14 × 14 matrices over GF(13).
The representations of 22.Sz(8).3 available are
• C and D as 24 × 24 matrices over GF(5).
• C and D as 120 × 120 matrices over GF(7).
• C and D as 48 × 48 matrices over GF(13).

### Maximal subgroups

The maximal subgroups of Sz(8) are
The maximal subgroups of Sz(8):3 are