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×#13244 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #466 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,24,31,26,32,25,50)(20,33,37,53,38,61,49,64)(21,39,42,71,44,72,43,58)(22,30,23,48,47,62,46,29)(27,51,54,35,56,36,55,45)(28,57,59,74,60,40,70,41)(34,52,67,77,68,79,75,78)(63,73,69,76,80,65,81,66);(14,15,16,17)(18,20,22,45,25,37,46,51,26,38,47,35,24,49,23,36)(19,27,48,33,31,55,62,64,32,56,29,61,50,54,30,53)(21,34,40,73,43,67,74,76,44,68,57,65,42,75,41,66)(28,52,71,63,59,78,72,81,60,79,58,80,70,77,39,69);(18,21,22,40,25,43,46,74,26,44,47,57,24,42,23,41)(19,28,48,71,31,59,62,72,32,60,29,58,50,70,30,39)(20,34,45,73,37,67,51,76,38,68,35,65,49,75,36,66)(27,52,33,63,55,78,64,81,56,79,61,80,54,77,53,69)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6232);