ATLAS: HigmanSims group HS
Order = 44352000 = 2^{9}.3^{2}.5^{3}.7.11.
Mult = 2.
Out = 2.
The following information is available for HS:
Standard generators of the HigmanSims 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 2.HS are preimages 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 either 2.HS.2 are preimages 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.
Finding generators
To find standard generators for HS:
 Find any element of order 20. This powers up to a 2Aelement x
and a 5Aelement y.
 Find a conjugate a of x and a conjugate b of
y, whose product has order 11.
This algorithm is available in computer readable format:
finder for HS.
To find standard generators for HS.2:
 Find any element of order 30 and call it y. Its 15th power,
x say, is a 2Celement.
[The probability of success at each attempt is 1 in 30 (or 1 in 15 if you
consider outer elements only).]
 Find conjugates c of x and z of y such that
cz has order 5 and czzz has order 20.
[The probability of success at each attempt is 3 in 110 (about 1 in 37).]
 Now c and d = cz are standard generators of HS:2.
This algorithm is available in computer readable format:
finder for HS.2.
Checking generators
To check that elements x and y of HS
are standard generators:
 Check o(x) = 2
 Check o(y) = 5
 Check o(xy) = 11
 Check o(xyy) = 10
 Check o(xyxyy) = 15
This algorithm is available in computer readable format:
checker for HS.
To check that elements x and y of HS.2
are standard generators:
 Check o(x) = 2
 Check o(y) = 5
 Check o(xy) = 30
 Check o([x,y]) = 3
This algorithm is available in computer readable format:
checker for HS.2.
Presentations of HS and HS:2 in terms of their standard generators are
given below.
< a, b  a^{2} = b^{5} =
(ab)^{11} = (ab^{2})^{10} =
[a, b]^{5} =
[a, b^{2}]^{6} =
[a, bab]^{3} =
ababab^{2}ab^{1}ab^{2}ab^{1}ab^{2}abab(ab^{2})^{4} =
ab(ab^{2}(ab^{2})^{2})^{2}ab^{2}abab^{2}(ab^{1}ab^{2})^{2} =
abab(ab^{2})^{2}ab(ab^{1})^{2}ab(ab^{2})^{2}ababab^{2}ab^{1}ab^{2}
= 1 >.
< c, d  c^{2} = d^{5} =
[c, d]^{3} =
[c, d^{2}]^{4} =
((cd)^{4}cd^{2}cd^{1}cd^{1}cd^{2})^{2} =
[c, dcdcd^{2}cd^{2}(cd^{2})^{2}cd^{1}] =
[c, dcdcd^{2}cd^{1}(cd^{2})^{3}] =
[c, d^{1}cd^{1}cd^{2}cd(cd^{2})^{3}]
= 1 >.
These presentations are available in Magma format as follows:
HS on a and b and
HS:2 on c and d.
The representations of HS available are:
 Some primitive permutation representations.

Permutations on 100 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 176b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1100a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 1100b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 3850 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 4125 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 5600 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 15400 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 2modular representations.

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 56 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 132 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 518 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 1000 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 3modular representations.

Dimension 22 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 49 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 321 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 693 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 748 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 825 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 5modular representations.

Dimension 21 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 55 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 98 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
NB: The above two representations have been swapped relative to Version 1.

Dimension 175 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 210 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 280 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 518 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 650 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 7modular representations.

Dimension 22 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 175 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 605 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 693 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 803 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(49):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some 11modular representations

Dimension 22 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 77 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 174 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 231 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 693 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 770 over GF(121):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 825 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 854 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(11):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.HS available are:

Permutations on 704 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 4400 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 11200 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Some 3modular representations.

Dimension 56 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 176 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 440 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Some 5modular representations.

Dimension 28 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 120 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 440 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(11):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of HS:2 available are:

Permutations on 100 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 352 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 1100b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 15400 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All irreducibles over GF(2):

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 56 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 132 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 518 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1000 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1408 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 1792 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 22 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 a reducible representation obtained from the Leech lattice.
It is the uniserial module of shape 20.1.1 for HS:2.
 Some irreducibles over GF(3):

Dimension 22 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 77 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 98 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 308 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 321 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 693 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 748 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 825 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles over GF(5):

Dimension 21 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 55 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 98 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 175 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 210 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 266 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 518 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 560 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 650 over GF(5):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles over GF(7):

Dimension 22 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 77 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 175 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 308 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 605 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 693 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 770 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 803 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles over GF(11):

Dimension 22 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 77 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 174 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 231 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 308 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 693 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 770 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 825 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 854 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 896 over GF(11):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 22 over Z:
c and d (Magma).
The representations of 2.HS:2 available are:

Permutations on 1408 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 112 over GF(3)  reducible over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(9):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 56 over GF(5):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of the HigmanSims group are as follows.
Words provided by Suleiman and Wilson.
 M22, with standard generators
a, (abb)^5(bababb)^2(abb)^5.
 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 × A6.2.2, with three generators
here.
 5:4 × 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.
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.
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old HS page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 16th June 2000.
Last updated 6.1.05 by SJN.
Information checked to
Level 0 on 16.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.