Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#33458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator | C8 | C23×C4×C13 |
| Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C23×C13 | C2×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ C13×#405 ◁ G | |
646 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(18,19)(20,30)(21,26)(22,38)(23,43)(24,27)(25,32)(28,41)(29,40)(31,55)(33,62)(34,56)(35,50)(36,65)(37,68)(39,53)(42,54)(44,52)(45,58)(46,61)(47,70)(48,67)(49,64)(51,60)(57,77)(59,80)(63,76)(66,75)(69,73)(71,81)(72,78)(74,79);(18,20,24,34)(19,25,27,46)(21,31,35,57)(22,39,42,70)(23,44,40,33)(26,45,50,72)(28,52,51,62)(29,53,43,47)(30,54,56,38)(32,60,61,41)(36,66,63,81)(37,69,64,59)(48,73,74,80)(49,75,68,71)(55,76,77,65)(58,79,78,67);(14,16,17,15)(18,21,24,35)(19,26,27,50)(20,31,34,57)(22,36,42,63)(23,37,40,64)(25,45,46,72)(28,48,51,74)(29,49,43,68)(30,55,56,77)(32,58,61,78)(33,59,44,69)(38,65,54,76)(39,66,70,81)(41,67,60,79)(47,71,53,75)(52,73,62,80);(19,27)(21,35)(25,46)(28,51)(29,43)(30,56)(31,57)(36,63)(37,64)(38,54)(47,53)(52,62)(58,78)(59,69)(66,81)(67,79)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26446);