Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#33789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ C13×#405 ◁ G | |
562 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)(16,20)(17,19)(18,21)(22,23,28,33)(24,29,35,50)(25,38,41,72)(26,42,46,71)(27,47,44,32)(30,53,55,39)(31,56,58,45)(34,60,61,83)(36,66,64,82)(37,69,65,52)(40,73,48,57)(43,59,74,54)(49,75,76,62)(51,79,78,68)(63,84,70,80)(67,81,85,77);(14,16,17,21)(15,18,19,20)(22,24)(23,29)(25,34)(26,36)(27,37)(28,35)(30,49)(31,51)(32,52)(33,50)(38,60)(39,62)(40,63)(41,61)(42,66)(43,67)(44,65)(45,68)(46,64)(47,69)(48,70)(53,75)(54,77)(55,76)(56,79)(57,80)(58,78)(59,81)(71,82)(72,83)(73,84)(74,85);(16,21)(18,20)(22,25,28,41)(23,30,33,55)(24,34,35,61)(26,43,46,74)(27,48,44,40)(29,49,50,76)(31,57,58,73)(32,59,47,54)(36,67,64,85)(37,70,65,63)(38,71,72,42)(39,56,53,45)(51,80,78,84)(52,81,69,77)(60,82,83,66)(62,79,75,68);(15,19)(18,20)(24,35)(29,50)(34,61)(36,64)(37,65)(49,76)(51,78)(52,69)(60,83)(62,75)(63,70)(66,82)(67,85)(68,79)(77,81)(80,84)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26777);