ATLAS: Harada–Norton group HN

Order = 273030912000000 = 2¹⁴.3⁶.5⁶.7.11.19.
Mult = 1.
Out = 2.

The following information is available for HN:

Standard generators.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of the Harada–Norton group HN are a and b where a is in class 2A, b is in class 3B, ab has order 22 and ababb has order 5.
Standard generators of its automorphism group HN:2 are c and d where c is in class 2C, d is in class 5A and cd has order 42.

A pair of elements conjugate to (a, b) may be obtained as
a' = (cd)^{-3}(cdcdcdcddcdcddcdd)^{10}(cd)^3, b' = (cdd)^{8}(cdcdd)^5(cdd)^{10}.

The outer automorphism may be realised by mapping (a,b) to (a,(abb)^-8b(abb)^8).

Black box algorithms

Finding generators

To find standard generators for HN:

Find any element of order 14 or 22. It powers up to a 2A-element, x say.
Find any element of order 9. This powers up to a 3B-element, y say.
Find a conjugate a of x and a conjugate b of y whose product has order 22 and whose commutator has order 5.

This algorithm is available in computer readable format: finder for HN.

To find standard generators for HN.2:

Find any element of order 18 or 42. It powers up to a 2C-element, x say.
Find any element of order 35 or 60. This powers up to a 5A-element, y say.
Find a conjugate a of x and a conjugate b of y, whose product has order 42.

This algorithm is available in computer readable format: finder for HN.2.

Checking generators

To check that elements x and y of HN are standard generators:

Check o(x) = 2
Check o(y) = 3
Check o(xy) = 22
Check o(xyxyy) = 5
Check o(x((xy)¹¹)^xyyxyxyxyxyy) = 5

This algorithm is available in computer readable format: checker for HN.

To check that elements x and y of HN.2 are standard generators:

Check o(x) = 2
Check o(y) = 5
Check o(xy) = 42
Let z = xy³(xy)⁴
Check o(z) = 60
Let t = z³⁰
Check o(ty) = 22
Check o(ty²(ty)³) = 22

This algorithm is available in computer readable format: checker for HN.2.

Representations

The representations of HN available are:

Permutations on 1140000 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Some representations in characteristic 2.
- Dimension 132 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
- Dimension 132 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
- Dimension 133 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — indecomposable with constituents 132.1.
- Dimension 760 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
- Dimension 2650 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 133 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 133 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 760 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — kindly provided by Jürgen Müller.
Dimension 133 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 626 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 627 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). — uniserial 626.1.
Dimension 133 over GF(49): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 133 over GF(49): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 760 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 133 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 133 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 760 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 133 over GF(19): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 133 over GF(19): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
Dimension 760 over GF(19): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

The representations of HN:2 available are:

Dimension 264 over GF(2): 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).
Permutations on 1140000 points: a and b (Meataxe binary).

Maximal subgroups

The maximal subgroups of HN are:

A12, with generators here.
2.HS.2, with standard generators (abababb)^-1bababb, (abababbab)^-2(ababb)(abababbab)^2.
U3(8):3, with generators here.
2^1+8.(A5 × A5).2, with generators here.
(D10 × U3(5)).2, with generators here.
5^1+4.2^1+4.5.4, with generators here.
2^6.U4(2), with generators here.
(A6 × A6).D8
2^3+2+6.(3 × L3(2))
5^2+1+2.4.A5, with generators here.
M12:2, with generators here.
M12:2, with generators here.
3^4:2.(A4 × A4).4
3^1+4:4.A5, with generators here.

The maximal subgroups of HN:2 are:

HN, with standard generators (cd)^{-3}(cdcdcdcddcdcddcdd)^{10}(cd)^3, (cdd)^{8}(cdcdd)^5(cdd)^{10}.
S12 with generators here.
4.HS.2 with generators here.
U3(8):6 with generators here.
2^1+8.(A5 × A5).2.2 with generators here.
5:4 × U3(5):2 with generators here.
5^1+4.2^1+4.5.4.2 with generators here.
2^6.U4(2).2 with generators here.
(S6 x S6).2.2 with generators here.
2^3+2+6.(3 × L3(2)).2 with generators here.
5^2+1+2.4.A5.2 with generators here.
3^4:2.(S4 × S4).2 with generators here.
3^1+4:4.S5 with generators here.

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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Version 2.0 created on 14th June 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 14.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.