ATLAS: Harada-Norton group HN

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

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}.

Black box algorithms

To find standard generators for HN: To find standard generators for HN.2:


The representations of HN available are The representations of HN:2 available are

Maximal subgroups

The maximal subgroups of HN are The maximal subgroups of HN: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 1.12.99

R.A.Wilson, R.A.Parker and J.N.Bray.