Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#13146 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1934 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
312 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)(14,15,19,25,21,26,20,43)(16,27,30,55,32,56,31,65)(17,33,36,68,38,50,37,70)(18,39,40,45,42,71,41,24)(22,44,47,64,48,63,54,29)(23,49,51,34,52,69,67,35)(28,57,60,73,62,75,61,46)(53,74,76,58,66,72,77,59);(14,16,18,29,19,30,40,44,21,32,42,64,20,31,41,63)(15,22,45,56,43,54,39,55,26,48,24,27,25,47,71,65)(17,28,34,58,36,60,69,72,38,62,35,59,37,61,49,74)(23,46,33,66,67,75,70,76,52,73,50,53,51,57,68,77);(14,17,18,34,19,36,40,69,21,38,42,35,20,37,41,49)(15,23,24,50,43,67,71,68,26,52,45,33,25,51,39,70)(16,28,29,58,30,60,44,72,32,62,64,59,31,61,63,74)(22,46,27,53,54,75,65,77,48,73,56,66,47,57,55,76)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6134);