# ATLAS: Unitary group U3(5)

Order = 126000.
Mult = 3.
Out = S3.

### Standard generators

Standard generators of U3(5) are a and b where a has order 3, b is in class 5A, and ab has order 7.
Standard generators of 3.U3(5) are pre-images A and B where B has order 5, and AB has order 7.
Standard generators of U3(5):2 are c and d where c is in class 2B, d is in class 4A, cd has order 10, and cdcdddcdd has order 2.
Standard generators of 3.U3(5):2 are pre-images C and D where D has order 4.

### Representations

The representations of U3(5) available are
• a and b as permutations on 50 points.
• All faithful irreducibles in characteristic 2.
• a and b as 20 x 20 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 28 x 28 matrices over GF(2).
• a and b as 104 x 104 matrices over GF(2).
• a and b as 144 x 144 matrices over GF(2).
• a and b as 144 x 144 matrices over GF(2).
• All faithful irreducibles in characteristic 3.
• a and b as 20 x 20 matrices over GF(3).
• a and b as 21 x 21 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 28 x 28 matrices over GF(3).
• a and b as 84 x 84 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 126 x 126 matrices over GF(3).
• a and b as 144 x 144 matrices over GF(9).
• a and b as 144 x 144 matrices over GF(9).
• All faithful irreducibles in characteristic 5.
• a and b as 8 x 8 matrices over GF(5).
• a and b as 10 x 10 matrices over GF(25).
• a and b as 10 x 10 matrices over GF(25).
• a and b as 19 x 19 matrices over GF(5).
• a and b as 35 x 35 matrices over GF(25).
• a and b as 35 x 35 matrices over GF(25).
• a and b as 63 x 63 matrices over GF(5).
• a and b as 125 x 125 matrices over GF(5) - the Steinberg representation.
• All faithful irreducibles in characteristic 7.
• a and b as 20 x 20 matrices over GF(7).
• a and b as 21 x 21 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 28 x 28 matrices over GF(7).
• a and b as 84 x 84 matrices over GF(7).
• a and b as 105 x 105 matrices over GF(7).
• a and b as 124 x 124 matrices over GF(7).
• a and b as 126 x 126 matrices over GF(7).
• a and b as 126 x 126 matrices over GF(49).
• a and b as 126 x 126 matrices over GF(49).
The representations of U3(5):2 available are
• c and d as 8 x 8 matrices over GF(5).
• c and d as permutations on 50 points.
• c and d as permutations on 126 points.
• c and d as permutations on 175 points.
The representations of 3.U3(5) available are
• A and B as 3 x 3 matrices over GF(25) - the natural representation.
The representations of 3.U3(5):2 available are
• C and D as 6 x 6 matrices over GF(5).

### Maximal subgroups

The maximal subgroups of U3(5) are as follows.
The maximal subgroups of U3(5):2 are as follows.
The maximal subgroups of U3(5):3 are as follows.
• U3(5)
• 6^2:S3
• 7:3 x 3
• 5^1+2:24
• 3^2:2A4
• 2.S5 x 3
The maximal subgroups of U3(5):S3 are as follows.
• U3(5):3
• U3(5):2
• 6^2:D12
• (7:3 x 3):2
• 5^1+2:24:2
• 3^2:2S4
• (3 x 2.S5).2

### Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy have been dealt with.
We have made a choice of the cyclic ordering of classes 5B/C/D, which may change if we ever decide on a consistent convention for such things.