# ATLAS: Unitary group U3(3), Derived group G2(2)'

Order = 6048 = 25.33.7.
Mult = 1.
Out = 2.

### Standard generators

Standard generators of U3(3) are a and b where a has order 2, b has order 6 and ab has order 7.

Standard generators of U3(3):2 = G2(2) are c and d where c is in class 2B, d is in class 4D and cd has order 7.

### Presentations

Presentations of U3(3) and U3(3):2 = G2(2) on their standard generators are given below.

< a, b | a2 = b6 = (ab)7 = [a, (ab2)3] = b3[b2, ab3a]2 = 1 >.

< c, d | c2 = d4 = (cd)7 = [c, d]6 = (cd(cd2)3)2 = [d2, cdc]3 = 1 >.

### Representations

The representations of U3(3) available are:
• All primitive permutation representations.
• Permutations on 28 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 63[a] points - on the cosets of 4.S4: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 63[b] points - on the cosets of 4^2:S3: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 2.
• Dimension 6 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 14 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32[a] over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 32[b] over GF(2) - the dual of the above: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• Dimension 3[b] over GF(9) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 6[b] over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 7 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 15[b] over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 27 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
• Some faithful irreducibles in characteristic 0
• Dimension 6 over Z[i]: a and b (GAP).
• Dimension 7 over Z: a and b (GAP).
• Dimension 7 over Z[i]: a and b (GAP).
• Dimension 14 over Z (reducible over Z[i]): a and b (GAP).
• Dimension 14 over Z: a and b (GAP).
• Dimension 21 over Z: a and b (GAP).
• Dimension 21 over Z[i]: a and b (GAP).
• Dimension 42 over Z (reducible over Z[i]): a and b (GAP).
• Dimension 27 over Z: a and b (GAP).
• Dimension 28 over Z[i]: a and b (GAP).
• Dimension 56 over Z (reducible over Z[i]): a and b (GAP).
• Dimension 64 over Z (reducible over Q(b7)): a and b (GAP).
The representations of U3(3):2 = G2(2) available are:
• Permutations on 63 points - on the cosets of 4^2:D12: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 2.
• Dimension 6 over GF(2) - exhibiting the isomorphism with G2(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 64 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 3 - up to tensoring with linear characters.

### Maximal subgroups

The maximal subgroups of U3(3) are as follows.
• 31+2:8, with generators here.
• L2(7), with generators here.
• 4.S4.
• 42:S3, with generators here.
The maximal subgroups of U3(3):2 = G2(2) are as follows.
• U3(3), with standard generators dd, cdcddd.
• 31+2:8:2.
• L2(7):2.
• 4.S4:2.
• 42:D12.

### Conjugacy classes

At the moment, we are only going to give enough class representatives so that we can sort out some of our generality problems.

Representatives of some of the 14 conjugacy classes of U3(3) are given below.

• 1A: identity [or a2].
• 2A: a.
• 3A: b2.
• 3B: abab-1 or [a, b].
• 4A: (ab-2)3.
• 4B: (ab2)3.
• 4C: ab3.
• 6A: b.
• 7A: ab.
• 7B: ab-1.
• 8A: ab2ab-1.
• 8B: abab-2.
• 12A: ab2.
• 12B: ab-2.
Representatives of some of the 16 conjugacy classes of U3(3):2 are given below.
• 1A: identity [or c2].
• 2A: d2.
• 3A: .
• 3B: .
• 4AB: .
• 4C: .
• 6A: .
• 7AB: cd.
• 8AB: .
• 12AB: .
• 2B: c.
• 4C: d.
• 6B: .
• 8C: .
• 12C: .
• 12D: . Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old U3(3) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 19th July 2000.
Last updated 03.03.04 by SJN.
Information checked to Level 0 on 19.07.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.