Order = 126000 = 24.32.53.7.
Mult = 3.
Out = S3.

Porting notes

Fully copied from version 2. 30/7/06.

Standard generators

Standard generators of U3(5) are a, b where a has order 3, b is in class 5A and ab has order 7.

Standard generators of 3.U3(5) are preimages A, B where B has order 5 and AB has order 7.

Standard generators of U3(5):2 are c, 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 preimages C, D where D has order 4.

Standard generators of U3(5):3 are e, f where e is in class 2A, f is in class 3C/C' and ef has order 15.

Standard generators of 3.U3(5).3 are preimages E, F where E has order 2.

Standard generators of U3(5):S3 are g, h where g is in class 2B, h is in class 3C, gh has order 8 and ghghh has order 12.

Standard generators of 3.U3(5).S3 are preimages G, H.

Presentations

Group Presentation Link
U3(5) 〈 a, b | a3 = b5 = (ab)7 = (ab−1)7 = aba−1b2aba−1bab2a−1b = 1 〉 Details
U3(5):2 〈 c, d | c2 = d4 = (cd)10 = (cdcd−1cd2)2 = [c, dcd]4 = (cdcdcdcd2)7 = 1 〉 Details
U3(5):3 〈 e, f | e2 = f3 = (ef)15 = [e, f]6 = [e, (ef)4(ef−1)3(ef)4] = [e, (ef)3(ef−1)6(ef)3] = 1 〉 Details
U3(5):S3 〈 g, h | g2 = h3 = (gh)8 = [g, h]12 = (ghghgh−1)6 = [g, hgh]6 = (ghghgh−1ghgh−1)10 = 1 〉 Details

Representations

Representations of U3(5)

Representations of 3.U3(5)

Representations of U3(5):2

Maximal subgroups

Maximal subgroups of U3(5)

Subgroup Order Index Programs/reps
A7
A7
A7
51+2:8
M10 = A6.2
M10 = A6.2
M10 = A6.2
2.S5 = 2.A5.2

Maximal subgroups of U3(5):2

Subgroup Order Index Programs/reps
U3(5) Program: Standard generators
S7 Program: Generators
Program: Standard generators
51+2:8:2 Program: Generators
A6.22 Program: Standard generators
2.S5.2 Program: Generators
L2(7):2 Program: Standard generators

Maximal subgroups of U3(5):3

Subgroup Order Index Programs/reps
U3(5)
51+2:24
2.S5 × 3
62:S3
32:2A4
7:3 × 3

Maximal subgroups of U3(5):S3

Subgroup Order Index Programs/reps
U3(5):3
U3(5):2
51+2:24:2
(3 × 2.S5).2
62:D12
32:2S4
(7:3 × 3):2