# ATLAS: Higman-Sims group HS

Order = 44352000 = 29.32.53.7.11.
Mult = 2.
Out = 2.

The following information is available for HS:

### 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 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.

### Black box algorithms

#### Finding generators

To find standard generators for HS:

• Find any element 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.
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 2C-element.
[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

Presentations of HS and HS:2 in terms of their standard generators are given below.

< a, b | a2 = b5 = (ab)11 = (ab2)10 = [a, b]5 = [a, b2]6 = [a, bab]3 = ababab2ab-1ab-2ab-1ab2abab(ab-2)4 = ab(ab2(ab-2)2)2ab2abab2(ab-1ab2)2 = abab(ab2)2ab(ab-1)2ab(ab2)2ababab-2ab-1ab-2 = 1 >.

< c, d | c2 = d5 = [c, d]3 = [c, d2]4 = ((cd)4cd-2cd-1cd-1cd-2)2 = [c, dcdcd2cd-2(cd2)2cd-1] = [c, dcdcd-2cd-1(cd-2)3] = [c, d-1cd-1cd2cd(cd2)3] = 1 >.

These presentations are available in Magma format as follows:
HS on a and b and HS:2 on c and d.

### Representations

The representations of HS available are:
• Some primitive permutation representations.
• Some 2-modular representations.
• Some 3-modular representations.
• Some 5-modular 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 7-modular representations.
• Some 11-modular representations
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 3-modular representations.
• Some 5-modular representations.
• 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 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):
• Some irreducibles over GF(5):
• Some irreducibles over GF(7):
• Some irreducibles over GF(11):
• 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).

### Maximal subgroups

The maximal subgroups of the Higman-Sims group are as follows. Words provided by Suleiman and Wilson.
The maximal subgroups of HS:2 are as follows.

### 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.
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