Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#11999 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | #106 |
| commutator | C8 | C22×C8×C13 |
| Frattini, cosocle | C4×C8 | C23×C13 |
| socle | C22×C13 | C2×#106 |
| prime radical | C23×C13 | #106 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ C2×C4×C8×C13 ◁ G | |
234 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)(14,15,16,18,17,19,20,21)(22,23)(24,33,36,47)(25,30)(26,39)(27,42)(28,32)(29,35,44,49)(31,40)(34,46,37,51)(38,45)(41,52)(43,48,53,50);(22,24,26,35,27,37,40,50,28,36,41,49,25,34,38,48)(23,29,31,33,30,43,39,46,32,44,45,47,42,53,52,51);(22,25,28,27)(23,30,32,42)(24,34,36,37)(26,38,41,40)(29,43,44,53)(31,39,45,52)(33,46,47,51)(35,48,49,50)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4987);