Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#29509 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C13×#376 ◁ G | |
676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 3328 |
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)(30,31,36,40)(32,37,38,44)(33,42,45,41)(34,39,35,43);(14,16,17,21)(15,18,19,20)(22,23)(24,27)(25,26)(28,29)(30,32)(31,37)(33,43)(34,42)(35,41)(36,38)(39,45)(40,44);(23,26)(27,29)(30,33,34,44,36,45,35,37)(31,38,43,41,40,32,39,42);(15,19)(18,20)(22,24,25,28)(23,27,26,29)(30,34,36,35)(31,39,40,43)(32,41,38,42)(33,44,45,37)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22497);