Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13267 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C8×C13 | C2×D4 |
| commutator | C2×C4 | C22×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C2×C22⋊C8 |
| prime radical | C23×C13 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,18)(19,21)(22,23,26,31,27,30,40,43,28,32,41,46,25,39,34,49)(24,33,37,48,38,44,51,52,35,47,50,53,36,29,42,45);(15,18)(19,20)(22,24,34,42,25,36,41,50,28,35,40,51,27,38,26,37)(23,29,43,48,39,47,31,45,32,44,49,53,30,33,46,52);(14,16,17,21)(15,19,18,20)(24,35)(29,44)(33,47)(36,38)(37,50)(42,51)(45,52)(48,53)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6255);