# ATLAS: Unitary group U5(2)

Order = 13685760 = 210.35.5.11.
Mult = 1.
Out = 2.

The following information is available for U5(2):

### Standard generators

Standard generators of U5(2) are a and b where a is in class 2A, b has order 5 and ab has order 11.
(Wlog we take ab in 11A.)

Standard generators of U5(2).2 are c and d where c has order 2 (so is in class 2C), d has order 4 (so is in class 4D), cd has order 11 and cdcdd has order 4.
(Wlog we take cdcdcddcdcdcdcddcdcdd in class 16B.)

### Automorphisms

An outer automorphism may be obtained by mapping (a, b) to (a, (abb)^-5b(abb)^5). Another outer automorphism may be obtained by mapping (a, b) to (a, bbbb).
Both these automorphisms have order 2.

We may obtain standard generators of U5(2):2 as c = u and d = u(ab2)2(ab3)3(ab2)2, where u is the automorphism u : (a, b) -> (a, b-1).

### Presentations

Presentations of U5(2) and U5(2):2 on their standard generators are given below.

< a, b | a2 = b5 = (ab)11 = [a, b]3 = [a, b2]3 = [a, bab]3 = [a, bab2]3 = 1 >.

< c, d | c2 = d4 = (cd)11 = (cd2)6 = (cdcd2)4 = [c, d]9 = (cd-1(cdcdcd-1)2cd2)2 = [d2, cdcdc]3 = 1 >.

These presentations are available in Magma format as follows: U5(2) on a and b and U5(2):2 on c and d.

### Representations

The representations of U5(2) available are:
• All primitive permutation representations.
• Some irreducibles in characteristic 2.
• Some irreducibles in characteristic 3.
• Some irreducibles in characteristic 5.
• Some irreducibles in characteristic 11.
• Dimension 44 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 55 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 119 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 176 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• a and b as 11 x 11 matrices over GF(121).
• Some irreducibles in characteristic 7 (not dividing the group order!).
• a and b as 10 x 10 matrices over GF(7).
• a and b as 11 x 11 matrices over GF(7).
• Some irreducibles in characteristic 0.
• Dimension 10 over Z[i]: a and b (Magma).
• Dimension 10 over Z[w]: a and b (Magma).
• Dimension 10 over Z[i2]: a and b (Magma).
• Dimension 11a over Z[w]: a and b (Magma).
• Dimension 11b over Z[w]: a and b (Magma).
• Dimension 20 over Z - reducible over C: a and b (Magma).
The representations of U5(2):2 available are:
• All faithful primitive permutation representations.
• All faithful irreducibles in characteristic 2.
• Dimension 10 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Some irreducibles in characteristic 0.
• Dimension 22 over Z: c and d (Magma).

### Maximal subgroups

The maximal subgroups of U5(2) are as follows.
The maximal subgroups of U5(2):2 are as follows.

### Conjugacy classes

Generators of the maximal cyclic subgroups of U5(2) are available here, and a program to obtain representatives of all conjugacy classes of U5(2) from these is available here.

Generators of the maximal cyclic subgroups of U5(2):2 are available here, and a program to obtain representatives of all conjugacy classes of U5(2):2 from these is available here. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U5(2) page - ATLAS version 1. Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 14th December 2001.
Last updated 10.11.03 by JNB.
Information checked to Level 0 on 14.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.