# ATLAS: Unitary group U4(2), Symplectic group S4(3)

Order = 25920 = 26.34.5.
Mult = 2.
Out = 2.

The following information is available for U4(2):

### Standard generators

Standard generators of U4(2) = S4(3) are a and b, where a in class 2A and b has order 5 and ab has order 9.
Standard generators of the double cover 2.U4(2) = Sp4(3) are preimages A and B where B has order 5 and AB has order 9.

Standard generators of U4(2):2 = S4(3):2 are c and d, where c in class 2C and d has order 9 and cd has order 10.
Standard generators of either of the double covers 2.U4(2):2 are preimages C and D where D has order 9.

### Automorphisms

An outer automorphism of U4(2) of order 2 may be obtained by mapping (a, b) to (a, bbbb).

### Presentations

Presentations of U4(2) and U4(2):2 in terms of their standard generators are given below.

< a, b | a2 = b5 = (ab)9 = [a, b]3 = [a, bab]2 = 1 >.

< c, d | c2 = d9 = (cd2)8 = [c, d2]2 = [c, d3cd3] = 1 >.

### Representations

The representations of U4(2) available are:
These representations are ordered [supposedly] with respect to ab being in class 9A.
• All primitive permutation permutation representations.
• Permutations on 27 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 36 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 40a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the cosets of N(3AB).
• Permutations on 40b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the cosets of 3^3:S4.
• Permutations on 45 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3.
• Dimension 5 over GF(3) - the natural representation as O5(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 14 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 25 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 81 over GF(3) - the Steinberg representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 5.
• Some faithful irreducibles in characteristic 0
The representations of 2.U4(2) available are:
• Permutations on 80 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 240 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 3.
• Dimension 4 over GF(3) - the natural representation as Sp4(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 16 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 40 over GF(3): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of U4(2):2 available are:
• Permutations on 27 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 36 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 40a points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - the cosets of N(3AB).
• Permutations on 40b points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - the cosets of 3^3:(S4 × 2).
• Permutations on 45 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 2.
• Dimension 8 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 6 over GF(2) - the representation as O6-(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 14 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 40 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 64 over GF(2) - the Steinberg representation: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 5 over GF(3) - the representation as O5(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 6 over Z: c and d (Magma); c and d (Magma). - lattices E6* and E6.
• Dimension 81 over Z: c and d (Magma).
The representations of 2.U4(2):2 (ATLAS-version) available are:

### Maximal subgroups

The maximal subgroups of U4(2) are as follows.
• 24:A5.
• S6 = A6:2.
• 31+2:2A4.
• 33:S4.
• 2.(A4 × A4).2.
The maximal subgroups of U4(2):2 are as follows.
• U4(2).
• 24:S5.
• S6 × 2.
• 31+2:2S4.
• 33:(S4 × 2).
• 2.(A4 × A4).2.2.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of U4(2) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of U4(2).2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U4(2) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 25th July 2000.
Last updated 16.12.04 by JNB.
Information checked to Level 0 on 25.07.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.