Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#13187 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| prime radical | C23×C13 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
294 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)(20,30,34,48)(21,36)(22,26)(23,41)(24,29)(25,32,43,31)(27,38)(28,37)(33,46,35,47)(39,44,49,42)(40,45);(18,20,23,33,24,34,37,35)(19,25,28,42,29,43,41,44)(21,31,22,39,38,32,40,49)(26,30,36,47,45,48,27,46);(14,15,16,17)(18,21,37,40,24,38,23,22)(19,26,41,27,29,45,28,36)(20,31,35,49,34,32,33,39)(25,30,44,46,43,48,42,47)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6175);