Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#13204 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C22×C13×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
514 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)(24,32)(25,37)(26,29)(27,40)(28,34)(30,33)(31,38)(35,44)(36,51)(39,52)(41,50)(42,53)(43,48)(45,47)(46,49);(19,21)(22,24,33,47,27,36,39,50,26,35,48,53,38,49,25,34)(23,28,37,46,31,42,43,44,29,41,52,51,40,45,30,32);(14,15,16,17)(23,29)(28,41)(30,43)(31,40)(32,44)(37,52)(42,45)(46,51)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6192);