Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29458 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×D4×C13 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C13×#376 ◁ G | |
660 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,19)(18,21)(30,31,36,39,35,41,34,40)(32,37,42,45,33,44,43,38);(14,16,17,21)(15,18,19,20)(22,23)(24,27)(25,26)(28,29)(30,32)(31,37)(33,35)(34,43)(36,42)(38,40)(39,45)(41,44);(23,26)(27,29)(30,33,35,32)(31,38,41,45)(34,43,36,42)(37,39,44,40);(15,19)(18,20)(22,24,25,28)(23,27,26,29)(30,34,35,36)(31,39,41,40)(32,42,33,43)(37,38,44,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22446);