ATLAS: Held group He

Order = 4030387200.
Mult = 1.
Out = 2.

Standard generators

Standard generators of the Held group He are a and b where a is in class 2A, b is in class 7C, and ab has order 17.
Standard generators of its automorphism group He:2 are c and d where c is in class 2B, d is in class 6C, and cd has order 30.
A pair of elements conjugate to (a,b) may be obtained as a' = (cdcdcddcdcddcd)^{14}, b' = (cdd)^{-2}(cdcdcdd)^3(cdd)^2.

The outer automorphism of He may be realised by mapping (a,b) to (a,(abb)-2b(abb)2).

Black box algorithms

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


The representations of He available are The representations of He:2 available are

Maximal subgroups

The maximal subgroups of He are The maximal subgroups of He: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 dealt with as follows.
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