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×#13207 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C2×D8 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 3328 |
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)(27,41)(28,35)(30,40)(31,38)(33,45)(34,47)(36,53)(39,42)(43,50)(44,49)(46,52)(48,51);(19,21)(22,24,33,49,27,36,25,35,26,34,40,52,38,51,39,50)(23,28,42,53,31,44,30,32,29,43,37,48,41,46,45,47);(14,15,16,17)(23,29)(24,34)(30,45)(31,41)(35,50)(36,51)(37,42)(49,52)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6195);