Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12825 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1802 |
| 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 | |
400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,18,26,21,27,20,24)(16,28,30,22,31,25,19,36)(17,32,34,43,35,45,33,41)(23,38,39,29,40,37,42,44);(14,16,18,30,21,31,20,19)(15,22,24,28,27,36,26,25)(17,29,33,38,35,44,34,37)(23,32,39,43,40,45,42,41);(14,17)(15,23)(16,29)(18,33)(19,37)(20,34)(21,35)(22,32)(24,39)(25,41)(26,42)(27,40)(28,43)(30,38)(31,44)(36,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5813);