# 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.