Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#33468 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #442 |
| commutator | C8 | C23×C4×C13 |
| Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C2×C13 | C2×#442 |
| prime radical | C13×#3414 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
652 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)(16,27,30,37)(17,31)(18,24)(19,23)(20,35)(21,26)(22,28,39,45)(25,33)(29,42,36,43)(32,41)(34,38,44,40);(14,16)(15,22)(17,28)(18,29)(19,34)(20,36)(21,30)(23,37)(24,38)(25,42)(26,39)(27,41)(31,43)(32,44)(33,45)(35,40);(14,17,20,19,21,33,18,32)(15,23,24,31,26,41,35,25)(16,28,36,34,30,45,29,44)(22,37,38,43,39,27,40,42);(14,18,21,20)(15,24,26,35)(16,29,30,36)(17,32,33,19)(22,38,39,40)(23,31,41,25)(27,42,37,43)(28,44,45,34)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26456);