Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62021 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#529 ◁ G | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(22,23)(24,29)(25,34)(26,31)(27,36)(28,32)(30,35)(33,37);(14,15)(16,19)(17,18)(20,21)(22,24,28,25)(23,29,32,34)(26,35,27,33)(30,36,37,31);(22,25,28,24)(23,30,32,37)(26,35,27,33)(29,36,34,31);(15,18)(19,21);(14,16,17,20)(15,19,18,21)(22,26,28,27)(23,31,32,36)(24,33,25,35)(29,37,34,30)637 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55009);