ATLAS: Suzuki group Sz(8)

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

The following information is available for Sz(8):


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, 7C, 7B) in that order, and thus is the Frobenius automorphism *4.
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 b is in class 4A, abb is in class 7A and ababbb is in class 13A (equivalently, abababbababbabb is in class 13B).

For Sz(8):3, we decree that d is in class 3A (so cd is in class 15A and cdcdcdd is in class 6A, etc) and cdcdcddcdcdd is in class 12A. (We also decree that class 12A' is class 12A*5; thus class 12A inverts into class 12B'.)


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. This automorphism is considered to reside in class 6A'.
We may obtain standard generators of Sz(8):3 as (c, d) = (a, (ba)2u2(ba)-2) where u is the above automorphism.

Presentations

Presentations of Sz(8) and Sz(8):3 on their standard generators are given below.

< a, b | a2 = b4 = (ab)5 = (ab2)7 = [a, b]13 = (abab-1ab2)7 = 1 >.

< c, d | c2 = d3 = (cd)15 = (cdcdcd-1)6 = [c, dcd-1cd-1(cdcdcd-1)2] = (cd)6(cd-1)4(cd)4cd-1(cd)3(cd-1)5 = 1 >.

These presentations, and those of some covering groups, are available in Magma format as follows:
Sz(8) on a and b; Sz(8) on a and b; 2.Sz(8) on A and B; Sz(8):3 on c and d; and 22.Sz(8):3 on C and D.


Representations

The representations of Sz(8) available are: 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!) The representations of Sz(8):3 available are: The representations of 22.Sz(8):3 available are:

Maximal subgroups

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

Conjugacy classes

A set of generators for the maximal cyclic subgroups of Sz(8) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. The conjugacy classes are available by running this program.

A set of generators for the maximal cyclic subgroups of Sz(8):3 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. The conjugacy classes are available by running this program.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Exceptional groups page Go to exceptional groups page.
Old Sz(8) page Go to old Sz(8) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 17th April 2000.
Last updated 22.04.04 by JNB.
Information checked to Level 0 on 18.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.