# ATLAS: Unitary group U5(3)

Order = 258190571520 = 211.310.5.7.61.
Mult = 1.
Out = 2.
The page for the group 310.U5(3) (non­split extension) is available here.

The following information is available for U5(3):

### Standard generators

Standard generators for U5(3) are a and b where a is in class 3A, b has order 5 and ab has order 16.
Standard generators for U5(3):2 are c and d where c is in class ??, d has order ? and cd has order ??.

NB: Class 3A is the class of transvections in U5(3).

### Automorphisms

An [outer] automorphism of U5(3) of order 2 can be obtained by mapping (a, b) to (a-1, b).

### Presentations

< a, b | a3 = b5 = [a, bab-1ab] = [a, b-2ab2] = (babababa-1)4 = (ababa-1bab2)5 = 1 >.

### Representations

The representations of U5(3) available are:
• Dimension 5[a] over GF(9) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 5[b] over GF(9) - dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10[a] over GF(9) - skew square of 5a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10[b] over GF(9) - skew square of 5b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 15[a] over GF(9) - symmetric square of 5a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 15[b] over GF(9) - symmetric square of 5b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 20 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24 over GF(3) - adjoint representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 30[a] over GF(9) - in 5a × 10a: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 30[b] over GF(9) - in 5b × 10b: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 30 over GF(3) - reducible over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 51 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).

### Maximal subgroups

The maximal subgroups of U5(3) are (I reckon) as follows [implementation of word programs not checked]: Go to main ATLAS (version 2.0) page. Go to classical groups page. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 24th September 2001.
Last updated 15.04.05 by RAW.
R.A.Wilson, S.J.Nickerson and J.N.Bray.