# ATLAS: Unitary group U3(8)

Order = 5515776 = 29.34.7.19.
Mult = 3.
Out = 3 × S3.

The following information is available for U3(8):

### Standard generators

Standard generators of U3(8) are a and b where a has order 2, b has order 3 (necessarily in class 3C) and ab has order 19.
Standard generators of 3.U3(8) = SU3(8) are preimages A and B where A has order 2 and AB has order 19.

Standard generators of U3(8):2 are c and d where c is in class 2B, d is in class 3C, cd has order 8 and cdcdcddcdcddcdd has order 9.
Standard generators of 3.U3(8):2 are preimages C and D where CDCDD has order 19.

Standard generators of U3(8):31 are e and f where e has order 2, f is in class 3D/E/F/D'/E'/F', ef has order 12, efeff has order 7 and efefeffefeffeff has order 7.
These conditions distinguish classes 3D/E/F and 3D'/E'/F'.
Standard generators of 3.U3(8):31 are preimages E and F where E has order 2 and F has order 3.

Standard generators of U3(8):6 are g and h where g is in class 2B, h is in class 3D/D'/EF/EF' (i.e. an outer element of order 3), gh has order 18, ghghh has order 19 and ghghghhghghhghh has order 9.
These conditions force h to lie in a particular class of elements of order 3, and we label this class as 3D.
Standard generators of 3.U3(8):6 are preimages G and H where ...??... .
Standard generators for U3(8) may be obtained from those of U3(8):6 as (ghghghhgh)6, (gh)6.

Standard generators of U3(8):32 are i and j where i has order 2, j is in class 3G/G' and ij has order 9.
Standard generators of [any] 3.U3(8).32 are preimages I and J where I has order 2 and ...??... .

Standard generators of U3(8).33 are k and l where k has order 2, l is in class 9K/L/M/K'/L'/M', kl has order 9, kll has order 9, klll has order 6, kllll has order 18 and klkllkllll has order 9.
These conditions distinguish classes 9K/L/M and 9K'/L'/M'.

Standard generators of U3(8):S3 are m and n where m is in class 2B, n is in class 3G/G', mn has order 8, mnmnn has order 9 and (mn)3mn2mnmn2mn2 has order 14.
Standard generators of [any] 3.U3(8).S3 are preimages M and N where ...??... .

Standard generators of U3(8).32 are o and p where o is in class 3DEF/DEF', p is in class 9EFG/EFG', op has order 9, opp has order 9, oppp has order 12, opppp has order 9 and opoopp has order 7.
These conditions distinguish classes 3DEF and 3DEF'.

Standard generators of U3(8).(S3 × 3) are q and r where q is in class 2B, r is in class 9KLM/KLM', qr has order 6, qrqrr has order 3 and qrqrrqrrrr has order 6.
These conditions distinguish classes 9KLM and 9KLM'.

### Presentations

< a, b | a2 = b3 = (ab)19 = [a, b]9 = [a, bab]3 = (abababab-1)3ab-1ab(ab-1)3ab(ab-1)2 = (((ab)4(ab-1)3)2ab-1)2 = 1 >.

### Representations

The representations of U3(8) available are:
• Permutations on 513 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 3648 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• a and b as 8 x 8 matrices over GF(8).
• a and b as 27 x 27 matrices over GF(4).
• Dimension 56 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• a and b as 133 x 133 matrices over GF(3).
• a and b as 133 x 133 matrices over GF(3).
• a and b as 133 x 133 matrices over GF(3).
The representations of 3.U3(8) available are:
• Permutations on 4617 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 32832 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 3a over GF(64): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - the natural representation.
The representations of U3(8):2 available are:
The representations of U3(8):31 available are:
The representations of U3(8):6 available are:
• Permutations on 513 points: g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Permutations on 3648 points: g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Essentially all faithful irreducibles in characteristic 2.
• Dimension 24 over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 54a over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP). - restricting to U3(8) as 9a+9b+9c+9d+9e+9f.
• Dimension 54b over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP). - restricting to U3(8) as 27a+27b.
• Dimension 192 over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 432 over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 512 over GF(2): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 56 over GF(3): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 133 over GF(3): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
• Dimension 266 over GF(3): g and h (Meataxe), g and h (Meataxe binary), g and h (GAP).
The representations of U3(8):32 available are:
The representations of U3(8).33 available are:
The representations of U3(8):S3 available are:
The representations of U3(8).32 available are:
The representations of U3(8).(S3 × 3) available are:

### Conjugacy classes

• ab is in class 19A.
• (ABABABB)42 is the central element of order 3.
• cd is in class 8A.
• (ghghghhghghh)3 is in class 8A [forced by the choice for U3(8):2]. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U3(8) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 21st July 2004, from a version 1 file last updated on 25th May 2000.
Last updated 21.07.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.