# 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:
• Find any element of order 10 or 28. This then powers to a 2A-element x.
• Find any element of order 8. Its fourth power is a 2B-element z, say.
• Find two conjugates of z, whose product has order 7 or 21. This powers to a 7C-element y, say.
• Find a conjugate of x and a conjugate of y, whose product has order 17. These are standard generators of He.
To find standard generators for He.2:
• Find any element of order 16. Its eighth power is a 2B-element x, say.
• Find any element of order 30. Its fifth power is a 6C-element y, say
• Find a conjugate of x and a conjugate of y, whose product has order 30. These are standard generators of He.2.

### Representations

The representations of He available are
• a and b as 51 x 51 matrices over GF(2).
• a and b as 101 x 101 matrices over GF(2).
• a and b as 246 x 246 matrices over GF(2).
• a and b as 680 x 680 matrices over GF(2).
• a and b as 51 x 51 matrices over GF(9).
• a and b as 104 x 104 matrices over GF(5).
• a and b as 51 x 51 matrices over GF(25).
• a and b as 50 x 50 matrices over GF(7).
• a and b as 153 x 153 matrices over GF(7).
• a and b as 426 x 426 matrices over GF(7).
• a and b as 798 x 798 matrices over GF(7).
• a and b as 102 x 102 matrices over GF(17) - reducible over GF(289).
• a and b as permutations on 2058 points.
• a and b as permutations on 8330 points.
• a and b as permutations on 29155 points.
The representations of He:2 available are
• c and d as 102 x 102 matrices over GF(2).
• c and d as 102 x 102 matrices over GF(3).
• c and d as 102 x 102 matrices over GF(5).
• c and d as 104 x 104 matrices over GF(5).
• c and d as 50 x 50 matrices over GF(7).
• c and d as 153 x 153 matrices over GF(7).
• c and d as 426 x 426 matrices over GF(7).
• c and d as 102 x 102 matrices over GF(17).
• c and d as permutations on 2058 points.
• c and d as permutations on 8330 points.

### 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.
• The element 21AB is in class 21A, as it has trace 2 in the 104-dimensional representation over GF(5).
• The element 17AB is wlog in class 17A, independently of all other choices here.
• The elements 14CD, 21CD, and 28AB are linked already by power maps, and are wlog respectively 14C, 21D, and 28A.