Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13182 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #459 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1947 |
| prime radical | C23×C13 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
280 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)(14,15,20,25,21,26,19,36)(16,27,30,43,31,37,29,41)(17,24,18,34,33,42,32,23)(22,35,38,45,40,44,39,28);(14,16,20,30,21,31,19,29)(15,22,25,38,26,40,36,39)(17,28,18,35,33,45,32,44)(23,37,24,41,34,27,42,43);(14,17,19,32,21,33,20,18)(15,23,25,24,26,34,36,42)(16,28,29,44,31,45,30,35)(22,37,38,41,40,27,39,43)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6170);