Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#12994 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C24×C13 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C23×C4×C13 ◁ C23×C8×C13 ◁ G | |
530 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)(15,17)(18,19)(20,21)(22,23,27,32,42,51,40,49,28,33,26,31,25,30,41,50)(24,34,37,47,35,53,43,29,39,46,36,45,38,52,44,48);(14,15,16,17)(22,24)(23,29)(25,35)(26,43)(27,44)(28,39)(30,45)(31,52)(32,53)(33,48)(34,49)(36,40)(37,41)(38,42)(46,50)(47,51);(19,21)(22,25,28,42)(23,30,33,51)(24,35,39,38)(26,40,27,41)(29,45,48,47)(31,49,32,50)(34,53,46,52)(36,44,37,43)754 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5982);