ATLAS: Higman-Sims group HS
Order = 44352000 = 2^{9}.3^{2}.5^{3}.7.11.
Mult = 2.
Out = 2.
Standard generators
Standard generators of the Higman-Sims group HS are a
and b where
a is in class 2A,
b is in class 5A, and
ab has order 11.
Standard generators of the double cover 2HS are
pre-images A
and B where
B has order 5,
and AB has order 11.
Standard generators of the automorphism group HS:2 are
c
and d where
c is in class 2C,
d is in class 5C, and
cd has order 30.
Standard generators of 2HS:2 are
pre-images C
and D where
D has order 5.
A pair generators conjugate to
a, b can be obtained as
a' = (cd)^{-1}(cdd)^{10}cd,
b' = (cdcdd)^{-4}(cdd)^4(cdcdd)^4.
Black box algorithms
To find standard generators for HS:
- Find any element x of order 20. This powers up to a 2A-element x and a 5A-element y.
- Find a conjugate a of x and a conjugate b of y, whose product has order 11.
To find standard generators for HS.2:
- Find any element of order 14 or 30. It powers up to a 2C-element.
- I don't know how to find a 5C-element.
- Find a conjugate a of x and a conjugate b of y, whose product has order 30.
Representations
The representations of HS available are
- Some 2-modular representations
- a and
b as
20 x 20 matrices over GF(2).
- a and
b as
56 x 56 matrices over GF(2).
- a and
b as
132 x 132 matrices over GF(2).
- a and
b as
518 x 518 matrices over GF(2).
- a and
b as
1000 x 1000 matrices over GF(2).
- Some primitive permutation representations
- a and
b as
permutations on 100 points.
- a and
b as
permutations on 176 points.
- a and
b as
permutations on 1100 points - the cosets of L_{3}(4).2.
- a and
b as
permutations on 1100 points - the cosets of S8.
- a and
b as
permutations on 3850 points.
- a and
b as
permutations on 4125 points.
- a and
b as
permutations on 5600 points.
- a and
b as
permutations on 15400 points.
- Some 3-modular representations
- a and
b as
22 x 22 matrices over GF(3).
- a and
b as
49 x 49 matrices over GF(3).
- a and
b as
49 x 49 matrices over GF(3) - the image of the above under an outer
automorphism.
- a and
b as
77 x 77 matrices over GF(3).
- Some 5-modular representations
- a and
b as
21 x 21 matrices over GF(5).
- a and
b as
55 x 55 matrices over GF(5).
- a and
b as
98 x 98 matrices over GF(5).
- a and
b as
133 x 133 matrices over GF(5).
- a and
b as
133 x 133 matrices over GF(5) - the image of the above under an outer
automorphism.
- a and
b as
175 x 175 matrices over GF(5).
- a and
b as
210 x 210 matrices over GF(5).
- a and
b as
280 x 280 matrices over GF(5).
- Some 7-modular representations
- a and
b as
22 x 22 matrices over GF(7).
- Some 11-modular representations
- a and
b as
22 x 22 matrices over GF(11).
The representations of 2HS available are
- A and
B as
56 x 56 matrices over GF(3).
- A and
B as
176 x 176 matrices over GF(9).
- A and
B as
28 x 28 matrices over GF(5).
- A and
B as
120 x 120 matrices over GF(5).
- A and
B as
440 x 440 matrices over GF(5).
- A and
B as
permutations on 704 points.
The representations of HS:2 available are
- c and
d as
20 x 20 matrices over GF(2).
- c and
d as
22 x 22 matrices over GF(2) - a reducible representation obtained from
the Leech lattice.
- c and
d as
22 x 22 matrices over GF(3).
- c and
d as
21 x 21 matrices over GF(5).
- c and
d as
896 x 896 matrices over GF(11).
- c and
d as 22 × 22 matrices over Z.
- c and
d as
permutations on 100 points.
- c and
d as
permutations on 352 points.
- c and
d as
permutations on 1100 points - the cosets of S8 x 2.
- c and
d as
permutations on 15400 points.
The representations of 2HS:2 available are
- C and
D as
112 x 112 matrices over GF(3).
- C and
D as
56 x 56 matrices over GF(5).
- C and
D as
permutations on 1408 points.
Maximal subgroups
The maximal subgroups of the Higman-Sims group are as follows.
Words provided by Suleiman and Wilson.
- M22, with standard generators
a, (abb)^-5(bababb)^2(abb)^5.
(Corrected on 9/8/99.)
- U3(5):2,
with standard generators
(ababb)^5(ababababbababbabb)^3(ababb)^-5,
(ab)^4(abababbabababababbababb)(ab)^-4.
- U3(5):2
with standard generators
(ababb)^5(ababababbababbabb)^3(ababb)^-5,
(abb)^-2(abababbabababababbababb)(abb)^2.
- L3(4):2, with standard generators
(ab)^-2(ababababbababbabb)^3(ab)^2,
(abb)^-4(abababbabababababbababb)(abb)^4.
- S8, with standard generators
(ab)^-6(ababababbababbabb)^3(ab)^6,
(abb)^-7(abababb)(abb)^7.
- 2^4.S6, with generators
(mapping to standard generators of S6)
(abababbab)^5a(abababbab)^5,
(ab)^-3(babababbababb)(ab)^3.
- 4^3.L3(2), with generators
(mapping to standard generators of L3(2))
(ab)^-2(bababb)^2(ab)^2,
(abb)^-2(ababb)^5(abb)^2.
- M11, with standard generators
b^-1ab,
(abb)^-5(abababbabababababbababb)(abb)^5.
- M11, with standard generators
bab^-1,
(abbb)^-5((ab^-1)^2abbb(ab^-1)^4abbbab^-1abbb)(abbb)^5.
- 4.2^4.S5, with generators
(mapping to standard generators of S5)
a, a(a(abababbab)^5)^3.
- 2 x A6.2.2, with three generators
here.
- 5:4 x A5, with generators
(mapping to standard generators of A5)
(abababbab)^2(abb)^5(abababbab)^-2,
(ababb)^2b.
The maximal subgroups of HS:2 are as follows.
Words provided by Suleiman and Wilson.
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.
Return to main ATLAS page.
Last updated 09.08.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk