ATLAS: Suzuki group Sz(8)
Order = 29120 = 2^{6}.5.7.13.
Mult = 2^{2}.
Out = 3.
The following information is available for Sz(8):
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 2^{2}.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 2^{2}.Sz(8):3 are preimages
C
and D where
CDCDD has order 13.
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 2^{2}.Sz(8) are preimages
A
and B where
AB has order 5,
ABABB has order 7.
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'.)
An outer automorphism of Sz(8) of order 3 can be realised
by mapping (a, b) to
((ab)^{4}a(ab)^{4}, (abb)^{4}b(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)^{2}u^{2}(ba)^{2})
where u is the above automorphism.
Presentations of Sz(8) and Sz(8):3 on their standard generators are given below.
< a, b  a^{2} = b^{4} =
(ab)^{5} = (ab^{2})^{7} =
[a, b]^{13} =
(abab^{1}ab^{2})^{7} = 1 >.
< c, d  c^{2} = d^{3} =
(cd)^{15} = (cdcdcd^{1})^{6} =
[c, dcd^{1}cd^{1}(cdcdcd^{1})^{2}] =
(cd)^{6}(cd^{1})^{4}(cd)^{4}cd^{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
2^{2}.Sz(8):3 on C and D.
The representations of Sz(8) available are:

All primitive permutation representations.

Permutations on 65 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 520 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 imprimitive.

Permutations on 560 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1456 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 2080 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Essentially all faithful irreducibles in characteristic 2.

Dimension 4 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 16 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Steinberg representation.
 All faithful irreducibles in characteristic 5.

Dimension 14a over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14b over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35a over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35b over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35c over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 63 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65a over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65b over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65c over GF(125):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 195 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(125).
 Essentially all faithful irreducibles in characteristic 7.

Dimension 14 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 91 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 105 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 becomes 35abc over GF(343).
 All faithful irreducibles in characteristic 13.

Dimension 14 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 65 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 91 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
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!)

Permutations on 1040 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 128 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 8 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 40 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of Sz(8):3 available are:

All primitive permutation representations.

Permutations on 65 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 520 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 imprimitive.

Permutations on 560 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 1456 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 2080 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

All faithful irreducibles in characteristic 2 (up to tensoring with linear characters).

Dimension 12 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 48 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 64 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 63 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 105 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 195 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(49):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2^{2}.Sz(8):3 available are:

Permutations on 2080 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 24 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 120 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 48 over GF(13):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of Sz(8) are:
The maximal subgroups of Sz(8):3 are:

Sz(8), with standard generators
d^{1}cd,
(cdd)^{1}(cdcdcddcdcdd)^{3}cdd,
and standard generators
c,
(cdcddcdcddcd)^{3}._{ }

2^{3+3}:7:3, with generators
(cd)^{3}d(cd)^{3},
(cdd)^{1}cdcdcdcddcdcdd._{ }

13:12 = F_{156}, with generators
(cd)^{6}d(cd)^{6},
(cdd)^{7}(cdcdcddcdcdd)^{3}(cdd)^{7}.

5:4 × 3 = F_{20} × 3, with generators
(cd)^{3}c(cd)^{3},
(cdd)^{1}cdcdcddcdcddcdd.

7:6 = F_{42}, with generators
c, (cdd)^{1}dcdd
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.
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Version 2.0 created on 17th April 2000.
Last updated 15.04.05 by RAW.