ATLAS: Suzuki group Suz
Order = 448345497600.
Mult = 6.
Out = 2.
Standard generators and automorphisms
Standard generators of the Suzuki group Suz are a
and b where
a is in class 2B,
b is in class 3B,
ab has order 13,
and ababb has order 15.
Standard generators of 2.Suz are preimages A
and B where
B has order 3 and
AB has order 13.
Standard generators of 3.Suz are preimages A
and B where
A has order 2 and
AB has order 13.
Standard generators of 6.Suz are preimages A
and B where
A has order 4,
B has order 3 and
AB has order 13.
Standard generators of the automorphism group Suz:2 are c
and d where
c is in class 2C,
d is in class 3B, and
cd has order 28.
Standard generators of 2.Suz:2 are preimages C
and D where
D has order 3.
Standard generators of 3.Suz:2 are preimages C
and D where
D is in class +3B (equivalently,
CDCDD has order 7).
The outer automorphism of Suz may be realised by mapping
a, b to
(ab)^2bab, (abb)^2b(abb)^2.
If c' is the 15th power of this automorphism, and d' =b, then
(c',d') is conjugate to (c,d).
Black box algorithms
To find standard generators for Suz:
 Find any element of order 14. Its 7th power is a 2Belement.
 Find any element of order 9 or 18. This powers up to a 3Belement, y, say.
 Find a conjugate a of x and a conjugate b of y, whose product has order 13,
and whose commutator has order 15.
To find standard generators for Suz.2:
 Find any element of order 30. It powers up to a 2Celement.
 Find any element of order 9 or 18. This powers up to a 3Belement, y, say.
 Find a conjugate a of x and a conjugate b of y, whose product has order 28.
Representations
Representations are available for the following decorations of Suz:
The representations of Suz available are

a and
b as
permutations on 1782 points.

a and
b as
permutations on 22880 points.

a and
b as
permutations on 32760 points.
 a and
b as
110 x 110 matrices over GF(4).
 a and
b as
142 x 142 matrices over GF(2).
 a and
b as
64 x 64 matrices over GF(3).
 a and
b as
78 x 78 matrices over GF(3).
 a and
b as
286 x 286 matrices over GF(3).
 a and
b as
429 x 429 matrices over GF(3).
 a and
b as
649 x 649 matrices over GF(3).
 a and
b as
143 x 143 matrices over GF(5).
 a and
b as
143 x 143 matrices over GF(7).
 a and
b as
143 x 143 matrices over GF(11).
 a and
b as
143 x 143 matrices over GF(13).
The representations of 2.Suz available are

A and
B as
12 x 12 matrices over GF(3).

A and
B as
208 x 208 matrices over GF(3).

A and
B as
352 x 352 matrices over GF(3).
The representations of 3.Suz available are
 A and
B as
12 x 12 matrices over GF(4).
 A and
B as
66 x 66 matrices over GF(25).
The representations of 6.Suz available are
 A and
B as
12 x 12 matrices over GF(25).
 A and
B as
12 x 12 matrices over GF(7).
 A and
B as
12 x 12 matrices over GF(13).
The representations of Suz:2 available are
 c and
d as
permutations on 1782 points.
 c and
d as
64 x 64 matrices over GF(3).
 c and
d as
143 x 143 matrices over GF(5).
 c and
d as
143 x 143 matrices over GF(7).
 c and
d as
143 x 143 matrices over GF(11).
The representations of 2.Suz:2 available are
 C and
D as
12 x 12 matrices over GF(3).
The representations of 3.Suz:2 available are
 C and
D as
24 x 24 matrices over GF(2).
 C and
D as
132 x 132 matrices over GF(5).
 C and
D as
132 x 132 matrices over GF(7).
 C and
D as
132 x 132 matrices over GF(11).
 C and
D as
permutations on 5346 points.
The representations of 6.Suz:2 available are
 C and
D as
24 x 24 matrices over GF(3)  a reducible representation.
 C and
D as
24 x 24 matrices over GF(5).
 C and
D as
24 x 24 matrices over GF(11).
Maximal subgroups
The maximal subgroups of Suz are
 G2(4), with standard generators
(ab)^5(abababb)^6(ab)^5,
(abb)^4(ababb)^3(abb)^4.
 3.U4(3):2, with generators
b^1ab,
(abb)^4(abababababbababbabb)^2(abb)^4.
 U5(2), with standard generators
(abababbababb)^4,
(abababbab)^9(abababbabababababbababb)^6(abababbab)^9.
 2^1+6.U4(2), with generators
b^1ab,
(abb)^6(ababb)^3(abb)^6.
 3^5:M11, with generators (mapping to standard
generators of M11)
(ab)^6(abababbababb)^4(ab)^6,
(abb)^5(abababbababb)^6(abb)^5.
 J2:2, with generators
(ab)^4a(ab)^4,
(abb)^6(ababb)^3(abb)^6.
 2^4+6:3.A6, with generators
(ab)^5b(ab)^4, (abb)^6(abababbababb)^2(abb)^6.
 (A4 x L3(4)):2, with generators
here.
 2^2+8:(A5 x S3), with generators
here.
 M12:2, with standard generators
(ab)^2bab, (abb)^6b(abb)^6.
 3^2+4:2.(A4 x 2 x 2).2, with generators
here.
 (A6 x A5).2, with generators
here.
 (A6 x 3^2:4).2, with generators
here.
 L3(3):2, with standard generators here.
 L3(3):2, with standard generators here.
[NB old (nonstandard) generators for one of these L3(3):2 groups are
(ab)^6b(ab)^5, (abb)^3(abababbabababbababb)^2(abb)^3.]
 L2(25), with generators
a,
(ababababb)^1(ababb)^5(ababababb).
 A7, with (nonstandard) generators
(abababbab)^2a(abababbab)^2,
(ababababb)^5(abababababbababbabb)^2(ababababb)^5.
The maximal subgroups of Suz:2 are
Conjugacy classes
A set of generators for the maximal cyclic subgroups can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
Problems of algebraic conjugacy are not yet dealt with.
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Last updated 19.12.00
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk