Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#13339 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C13×#405 ◁ G | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 81 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 81 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19,25,31)(20,32,37,67)(21,38,44,57)(22,45,50,63)(23,30,24,29)(26,51,55,34)(27,40,60,41)(28,48,64,49)(33,52,71,80)(35,61,36,66)(39,76,70,56)(42,62,43,75)(46,53,73,54)(47,58,78,59)(65,68,81,69)(72,74,77,79);(14,15)(16,17)(18,20,25,37)(19,26,31,55)(21,39,44,70)(22,46,50,73)(23,36,24,35)(27,56,60,76)(28,61,64,66)(29,54,30,53)(32,63,67,45)(33,43,71,42)(34,48,51,49)(38,74,57,79)(40,68,41,69)(47,77,78,72)(52,59,80,58)(62,81,75,65);(15,17)(18,21)(19,27)(20,33)(22,47)(23,43)(24,42)(25,44)(26,52)(28,62)(29,59)(30,58)(31,60)(32,65)(34,72)(35,70)(36,39)(37,71)(38,45)(40,48)(41,49)(46,69)(50,78)(51,77)(53,76)(54,56)(55,80)(57,63)(61,74)(64,75)(66,79)(67,81)(68,73)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6327);