Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13636 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C2×D16 |
| commutator | C16 | C22×C4×C13 |
| Frattini, cosocle | C2×C16 | C23×C13 |
| socle | C2×C13 | C2×#366 |
| prime radical | C23×C13 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ C22×C13×C16 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,24,21,26,20,42)(16,27,31,53,29,51,33,64)(17,34,36,65,38,67,37,68)(19,40,39,50,41,69,35,25)(22,43,45,58,44,61,70,30)(23,47,48,57,49,56,66,28)(32,63,62,74,59,77,60,54)(46,55,52,71,72,75,73,76);(14,16,28,55,19,32,36,43,18,31,47,71,39,62,38,58,21,29,57,75,41,59,37,61,20,33,56,76,35,60,17,30)(15,22,34,54,25,46,66,64,42,70,68,77,69,73,49,51,26,44,67,74,50,72,48,53,24,45,65,63,40,52,23,27);(16,29)(22,44)(27,51)(30,58)(31,33)(32,59)(43,61)(45,70)(46,72)(52,73)(53,64)(54,74)(55,75)(60,62)(63,77)(71,76)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6624);