Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33687 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C23×C13 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ C13×#405 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 85 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23,26,31)(24,34,36,29)(25,39,40,75)(27,45,43,74)(28,48,44,33)(30,53,54,41)(32,57,56,47)(35,63,64,73)(37,69,67,78)(38,52,68,62)(42,76,49,58)(46,59,77,55)(50,65,60,82)(51,71,61,83)(66,80,72,84)(70,79,85,81);(15,18)(19,21)(22,24)(23,29)(25,35)(26,36)(27,37)(28,38)(30,50)(31,34)(32,51)(33,52)(39,73)(40,64)(41,65)(42,66)(43,67)(44,68)(45,78)(46,70)(47,71)(48,62)(49,72)(53,82)(54,60)(55,79)(56,61)(57,83)(58,80)(59,81)(63,75)(69,74)(76,84)(77,85);(22,25,26,40)(23,30,31,54)(24,35,36,64)(27,46,43,77)(28,49,44,42)(29,50,34,60)(32,58,56,76)(33,59,48,55)(37,70,67,85)(38,72,68,66)(39,74,75,45)(41,57,53,47)(51,80,61,84)(52,81,62,79)(63,78,73,69)(65,83,82,71);(14,16,17,20)(15,19,18,21)(24,36)(29,34)(35,64)(37,67)(38,68)(50,60)(51,61)(52,62)(63,73)(65,82)(66,72)(69,78)(70,85)(71,83)(79,81)(80,84)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26675);