# ATLAS: Held group He

Order = 4030387200 = 210.33.52.73.17.
Mult = 1.
Out = 2.

The following information is available for He:

### 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.

### Automorphisms

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

### Black box algorithms

#### Finding generators

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

#### Checking generators

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

• Check o(x) = 2
• Check o(y) = 7
• Check o(xy) = 17
• Check o(xyxyxyy) = 23
• Let z = xyyxyxyyxyy
• Check o(z) = 10
• Check o(xz5) = 3
This algorithm is available in computer readable format: checker for He.

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

• Check o(x) = 2
• Check o(y) = 6
• Check o(xy) = 30
• Let z = xyyxyyxy
• Check o(z) = 24
• Check o(xz12) = 17
• Let u = y3
• Let t = (u.ux)4((u.uxyyxy)2)
• Check o(t) = 15
• Check o([t,y]) = 1
This algorithm is available in computer readable format: checker for He.2.

### Presentations

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

< a, b | a2 = b7 = (ab)17 = [a, b]6 = [a, b3]5 = [a, babab-1abab] = (ab)4ab2ab-3ababab-1ab3ab-2ab2 = 1 >.

< c, d | c2 = d6 = ... = 1 > - we haven't found this one yet.

These presentations are available in Magma format as follows: He on a and b and .

### Representations

The representations of He available are:
• Some primitive permutation representations.
• Permutations on 2058 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 8330 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 29155 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 244800 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some irreducible representations in characteristic 2.
• Some irreducible representations in characteristic 3.
• Some irreducible representations in characteristic 5.
• Some irreducible representations in characteristic 7.
• Some irreducible representations in characteristic 17.
The representations of He:2 available are:
• Permutations on 2058 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 8330 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 202 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 492 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 306 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 679 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 104 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - Corrected (and renamed) on 6.10.04; previous representation generated 3.S7.
• Dimension 306 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(5): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 50 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 153 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 426 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 798 over GF(7): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 102 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 306 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 680 over GF(17): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).

### 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. Go to main ATLAS (version 2.0) page. Go to sporadic groups page. Go to old He page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 7th June 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 07.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.