# 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.
• a and b as permutations on 28 points.
• a and b as permutations on 36 points.
• a and b as permutations on 63 points (the cosets of 4.S4).
• a and b as permutations on 63 points (the cosets of 4^2:S3).
• All faithful irreducibles in characteristic 2.
• a and b as 6 × 6 matrices over GF(2).
• a and b as 14 × 14 matrices over GF(2).
• a and b as 32 × 32 matrices over GF(2).
• a and b as 32 × 32 matrices over GF(2) - the dual of the above.
• All faithful irreducibles in characteristic 3 (up to Frobenius automorphisms).
• a and b as 3 × 3 matrices over GF(9) - the natural representation.
• a and b as 6 × 6 matrices over GF(9).
• a and b as 7 × 7 matrices over GF(3).
• a and b as 15 × 15 matrices over GF(9).
• a and b as 27 × 27 matrices over GF(3).
• All faithful irreducibles in characteristic 7 (up to Frobenius automorphisms).
• a and b as 6 × 6 matrices over GF(7).
• a and b as 7 × 7 matrices over GF(7).
• a and b as 7 × 7 matrices over GF(49).
• a and b as 14 × 14 matrices over GF(7).
• a and b as 21 × 21 matrices over GF(7).
• a and b as 21 × 21 matrices over GF(49).
• a and b as 26 × 26 matrices over GF(7).
• a and b as 28 × 28 matrices over GF(49).
The representations of U3(3):2 = G2(2) available are:
• c and d as permutations on 63 points (the cosets of 4^2:D12).
• All faithful irreducibles in characteristic 2.
• c and d as 6 × 6 matrices over GF(2) - exhibiting the isomorphism with G2(2).
• c and d as 14 × 14 matrices over GF(2).
• c and d as 64 × 64 matrices over GF(2).
• All faithful irreducibles in characteristic 3 - up to tensoring with linear characters.
• c and d as 6 × 6 matrices over GF(3).
• c and d as 12 × 12 matrices over GF(3).
• c and d as 7 × 7 matrices over GF(3).
• c and d as 30 × 30 matrices over GF(3).
• c and d as 27 × 27 matrices over GF(3).

### Maximal subgroups

The maximal subgroups of U3(3) are as follows.
• 31+2:8.
• L2(7).
• 4.S4.
• 42:S3.
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.