# ATLAS: Unitary group U3(4)

Order = 62400 = 26.3.52.13.
Mult = 1.
Out = 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.
A generating outer automorphism may be obtained by mapping (a, b) to ((ab)^-2a(ab)^2, (abb)^-5b(abb)^5).

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.

### Presentations

< 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 >.

### Representations

The representations of U3(4) available are
• a and b as 3 × 3 matrices over GF(16) - the natural representation.
• a and b as 6 × 6 matrices over GF(4) - obtained from the above by regarding GF(16) as a 2-space over GF(4).
• a and b as 8 × 8 matrices over GF(4) - this one is absolutely irreducible.
• a and b as 16 × 16 matrices over GF(2) - the Steinberg representation.
• a and b as 12 × 12 matrices over GF(3).
• a and b as 64 × 64 matrices over GF(3).
• a and b as 12 × 12 matrices over GF(5).
• a and b as 39 × 39 matrices over GF(5).
• a and b as 65 × 65 matrices over GF(5).
• a and b as 39 × 39 matrices over GF(169).
• a and b as 63 × 63 matrices over GF(13).
• a and b as 65 × 65 matrices over GF(13).
• a and b as 208 x 208 matrices over GF(13) - really dimension 52 over GF(28561).
• a and b as permutations on 65 points.
• a and b as permutations on 208 points.
• a and b as permutations on 416 points.
The representations of U3(4):2 available are
• c and d as 6 × 6 matrices over GF(4).
The representations of U3(4):4 available are
NB: Changed to standard generators on 26.05.98.
• e and f as 12 × 12 matrices over GF(2).
• e and f as 12 × 12 matrices over GF(5).
• e and f as permutations on 65 points.
• e and f as permutations on 208 points.
• e and f as permutations on 416 points, on the cosets of the maximal subgroup 5^2:(4 × S3).
• e and f as permutations on 1600 points.

### Maximal subgroups

The maximal subgroups of U3(4) are as follows.
• 2^2+4:15
• 5 × A5
• 5^2:S3
• 13:3
The maximal subgroups of U3(4):2 are as follows.
• U3(4)
• 2^2+4:(3 × D10)
• D10 × A5
• 5^2:D12
• 13:6
The maximal subgroups of U3(4):4 are as follows.
• U3(4):2
• 2^2+4:(3 × D10).2
• (D10 × A5).2
• 5^2:(4 × S3)
• 13:12