Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12951 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1864 |
| 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 | |
406 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)(14,15,19,26,21,27,20,39)(16,28,31,40,33,44,32,43)(17,34,18,37,36,25,35,24)(22,30,41,29,42,45,23,38);(14,16)(15,22,27,42)(17,30)(18,38)(19,32)(20,31)(21,33)(23,39,41,26)(24,40,37,43)(25,44,34,28)(29,35)(36,45);(14,16)(15,23)(17,30)(18,38)(19,32)(20,31)(21,33)(22,39)(24,28)(25,40)(26,42)(27,41)(29,35)(34,43)(36,45)(37,44)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5939);