Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13140 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #481 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1926 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,20,26,18,24,21,36)(16,25,19,34,28,22,29,38)(17,30,31,41,33,45,32,37)(23,35,39,27,40,43,42,44);(14,16,21,29,18,28,20,19)(15,22,36,34,24,25,26,38)(17,27,31,43,33,44,32,35)(23,37,39,30,40,41,42,45);(14,17,20,32,18,33,21,31)(15,23,36,39,24,40,26,42)(16,27,19,35,28,44,29,43)(22,37,34,30,25,41,38,45)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6128);