Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19)(20,30)(21,34,36,41)(22,38)(23,25)(24,29)(26,32,44,48)(27,31)(28,33)(35,47,37,46)(39,43,49,45)(40,42);(14,15,16,17)(18,20,31,40,24,33,22,23)(19,25,38,28,29,42,27,30)(21,32,37,49,36,48,35,39)(26,41,45,46,44,34,43,47);(18,21)(19,26)(20,32)(22,35)(23,39)(24,36)(25,41)(27,43)(28,46)(29,44)(30,47)(31,37)(33,48)(34,42)(38,45)(40,49)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5812);