Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#33721 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| prime radical | C2×C13×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ C13×#405 ◁ G | |
794 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)(15,17)(18,19,24,29)(20,30,35,55)(21,26,36,47)(22,39,41,67)(23,42,40,28)(25,43,46,33)(27,50,51,31)(32,52,58,74)(34,62,61,54)(37,65,63,80)(38,66,64,49)(44,68,70,59)(45,56,53,69)(48,73,72,57)(60,79,78,76)(71,77,75,81);(18,20,31,56,23,34,41,43,24,35,50,69,40,61,22,33)(19,25,39,54,28,45,51,55,29,46,67,62,42,53,27,30)(21,32,57,77,38,60,63,68,36,58,73,81,64,78,37,59)(26,44,65,76,49,71,72,74,47,70,80,79,66,75,48,52);(14,15,16,17)(18,21)(19,26)(20,32)(22,37)(23,38)(24,36)(25,44)(27,48)(28,49)(29,47)(30,52)(31,57)(33,59)(34,60)(35,58)(39,65)(40,64)(41,63)(42,66)(43,68)(45,71)(46,70)(50,73)(51,72)(53,75)(54,76)(55,74)(56,77)(61,78)(62,79)(67,80)(69,81);(21,36)(26,47)(32,58)(37,63)(38,64)(44,70)(48,72)(49,66)(52,74)(57,73)(59,68)(60,78)(65,80)(71,75)(76,79)(77,81)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26709);