Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29475 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C2×C13×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C2×C13×#117 ◁ G | |
742 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)(17,18)(19,21)(22,23)(24,28)(25,27)(26,29)(30,31,36,39,35,40,34,41)(32,42,43,45,33,37,44,38);(15,18)(19,20)(22,24)(23,26)(25,29)(27,28)(30,32)(31,37)(33,35)(34,44)(36,43)(38,39)(40,42)(41,45);(24,29)(26,28)(30,33,35,32)(31,38,40,45)(34,44,36,43)(37,41,42,39);(14,16,17,21)(15,19,18,20)(23,27)(26,28)(30,34,35,36)(31,39,40,41)(32,43,33,44)(37,45,42,38)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22463);