Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C2×C13 | C2×#476 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
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)(16,28)(17,32,35,45)(18,24)(19,26)(20,36)(21,27)(22,30)(23,29,39,44)(25,31)(33,42,34,40)(37,41,43,38);(14,16,19,30,21,31,18,20)(15,22,24,28,27,36,26,25)(17,29,34,43,35,44,33,37)(23,32,38,42,39,45,41,40);(14,17)(15,23)(16,29)(18,33)(19,34)(20,37)(21,35)(22,32)(24,38)(25,40)(26,41)(27,39)(28,42)(30,43)(31,44)(36,45);(14,18,21,19)(15,24,27,26)(16,20,31,30)(17,33,35,34)(22,28,36,25)(23,38,39,41)(29,37,44,43)(32,42,45,40)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26625);