# ATLAS: Unitary group U3(4)

Order = 62400 = 26.3.52.13.
Mult = 1.
Out = 4.

The following information is available for U3(4):

### Standard generators

Standard generators of U3(4) are a and b where a has order 2, b has order 3 and ab has order 13.

Standard generators of U3(4):2 are c and d where c has order 2, d has order 3, cd has order 8, cdcdd has order 13 and cdcdcdcddcdcddcdd has order 10.
NB: Of course, c is in class 2B.

Standard generators of U3(4):4 are e and f where e is in class 2A, f is in class 4B or 4B', ef has order 12 and efefffeff has order 6.
NB: These conditions distinguish between classes 4B and 4B'. Classes 4B and 4B' are the classes for which the ATLAS class 4B is proxy.
With these conditions, f is conjugate to the *2 automorphism.

### Automorphisms

A generating outer automorphism of U3(4) may be obtained by mapping (a, b) to ((ab)-2a(ab)2, (ab-1)-5b(ab-1)5).
An outer automorphism of U3(4) of order 2 is given by mapping (a, b) to (a, b-1)

### Presentations

Presentations of U3(4), U3(4):2 and U3(4):4 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]5 = [a, babab]3 = 1 >.

< c, d | c2 = d3 = (cd)8 = [c, d]13 = [c, dcdcdcd-1cdcd]2 = [c, d-1cdcd]5 = 1 >.

< e, f | e2 = f4 = (ef)12 = [e, f]5 = (ef2)10 = efefef2efef2ef-1ef2efefef-1ef2ef-1ef2 = 1 >.

These presentations are available in Magma format as follows:
U3(4) on a and b, U3(4):2 on c and d and U3(4):4 on e and f.

### Representations

The representations of U3(4) available are:
• All primitive permutation representations.
• All faithful absolute irreducibles in characteristic 2.
• Dimension 3a over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 3b over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 3c over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 3d over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 8a over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 8b over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 9a over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 9b over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 9c over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 9d over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24a over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24b over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24c over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 24d over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 64 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the Steinberg representation.
• All other faithful irreducibles in characteristic 2.
• Dimension 6a over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 3 over GF(16).
• Dimension 6b over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 3 over GF(16).
• Dimension 12 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 3 over GF(16).
• Dimension 16 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 8 over GF(4).
• Dimension 18a over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 9 over GF(16).
• Dimension 18b over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 9 over GF(16).
• Dimension 36 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 9 over GF(16).
• Dimension 48a over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 24 over GF(16).
• Dimension 48b over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 24 over GF(16).
• Dimension 96 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 24 over GF(16).
• Essentially all faithful irreducibles in characteristic 3.
• Dimension 12 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 52e over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 13 over GF(81).
• Dimension 64 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 75a over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 75b over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 75c over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 75d over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 39 over GF(9).
• Dimension 208 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 52 over GF(81).
• Essentially all faithful irreducibles in characteristic 5.
• Dimension 12 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 39 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 65 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 150a over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 75 over GF(625).
• Dimension 150b over GF(25): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 75 over GF(625).
• Dimension 300 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 75 over GF(625).
• Essentially all faithful irreducibles in characteristic 13.
• Dimension 12 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 52e over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 13 over GF(28561).
• Dimension 63 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 65a over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 78 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 39 over GF(169).
• Dimension 208 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 52 over GF(28561).
• Dimension 260 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - really dimension 65 over GF(28561).
• a and b as 39 × 39 matrices over GF(169).
The representations of U3(4):2 available are:
• All primitive permutation representations.
• Permutations on 65 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 208 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 416 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - primitive.
• Permutations on 1600 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• c and d as 6 × 6 matrices over GF(4).
The representations of U3(4):4 available are:
• All primitive permutation representations.
• Permutations on 65 points: e and f (Meataxe), e and f (Meataxe binary), e and f (GAP).
• Permutations on 208 points: e and f (Meataxe), e and f (Meataxe binary), e and f (GAP).
• Permutations on 416 points: e and f (Meataxe), e and f (Meataxe binary), e and f (GAP). - on the cosets of the maximal subgroup 5^2:(4 × S3).
• Permutations on 1600 points: e and f (Meataxe), e and f (Meataxe binary), e and f (GAP).
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 with character in ABC.
• All faithful irreducibles in characteristic 5 with character in ABC.
• All faithful irreducibles in characteristic 13 with character in ABC.

### Maximal subgroups

The maximal subgroups of U3(4) are as follows.
The maximal subgroups of U3(4):2 are as follows.
The maximal subgroups of U3(4):4 are as follows.

NB: Maps between the various extensions of U3(4) have not been checked for compatibility with the class definitions (or even compatibility with each other).

### Conjugacy classes

Some conjugacy classes U3(4) are as follows.
• 1A: identity.
• 2A: a.
• 3A: b.
• 13A: ab.
Some conjugacy classes U3(4):2 are as follows.
• 1A: identity.
• 13AB: cdcdd. - compatible with U34d2G1-max1W1 and ab being in class 13A.
• 8A: cd.
• 8B: cdd.
Some conjugacy classes U3(4):4 are as follows.
• 1A: identity.
• 2A: e.
• 4B: f.
• 12A: ef.
• 16A: effefff.
• 16B: efeffeff.
• 4B': fff.
• 12A': efff.
• 16A': efeff.
• 16B': effeffefff.
• ab is in class 13A.
• cdcdd is in class 13AB.
• f is in class 4B (rather than 4B').
• Everything else should follow from definitions in the ABC (with consistency between the various extensions).

Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U3(4) page - ATLAS version 1.
Anonymous ftp access is also available. See here for details.

Version 2.0 created on 27th August 2004, from a version 1 file last updated on 12th April 2000.
Last updated 02.09.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.