Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#29510 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C22×C13×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C13×#376 ◁ G | |
758 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)(30,31,35,38)(32,36,40,43)(33,42,37,34)(39,45,44,41);(14,16,17,21)(15,18,19,20)(22,23)(24,27)(25,26)(28,29)(30,32,35,40)(31,36,38,43)(33,39,37,44)(34,41,42,45);(23,26)(27,29)(30,33,34,31,35,37,42,38)(32,39,41,36,40,44,45,43);(15,19)(18,20)(22,24,25,28)(23,27,26,29)(32,40)(36,43)(39,44)(41,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22498);