Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#33588 |
| 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 | |
680 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)(20,25,31,46)(21,34,37,68)(22,38,42,67)(23,43,40,28)(26,49,51,35)(27,52,54,41)(30,56,57,79)(32,62,60,78)(33,65,61,48)(36,69,44,53)(39,55,70,50)(45,71,72,58)(47,75,74,64)(59,80,66,76)(63,77,81,73);(14,15,16,17)(18,20)(19,25)(21,30)(22,32)(23,33)(24,31)(26,45)(27,47)(28,48)(29,46)(34,56)(35,58)(36,59)(37,57)(38,62)(39,63)(40,61)(41,64)(42,60)(43,65)(44,66)(49,71)(50,73)(51,72)(52,75)(53,76)(54,74)(55,77)(67,78)(68,79)(69,80)(70,81);(18,21,24,37)(19,26,29,51)(20,30,31,57)(22,39,42,70)(23,44,40,36)(25,45,46,72)(27,53,54,69)(28,55,43,50)(32,63,60,81)(33,66,61,59)(34,67,68,38)(35,52,49,41)(47,76,74,80)(48,77,65,73)(56,78,79,62)(58,75,71,64);(20,31)(25,46)(30,57)(32,60)(33,61)(45,72)(47,74)(48,65)(56,79)(58,71)(59,66)(62,78)(63,81)(64,75)(73,77)(76,80)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26576);