Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#26750 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C24×C13 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C22⋊C4 ◁ G | |
736 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)(16,19)(17,18)(20,21)(22,23)(24,27)(25,26)(28,29)(30,31,34,37)(32,39,42,36)(33,43,35,38)(40,44,41,45);(15,18)(19,21)(30,32,34,42)(31,36,37,39)(33,41,35,40)(38,45,43,44);(14,16,17,20)(15,19,18,21)(23,26)(27,29)(31,37)(32,40)(33,35)(36,44)(39,45)(41,42);(15,18)(19,21)(22,24,25,28)(23,27,26,29)(30,33,34,35)(31,38,37,43)(32,41,42,40)(36,45,39,44)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 19738);