Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#11498 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C22×C13 | C2×C22⋊C8 |
| prime radical | C23×C13 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 61 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 61 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,22,24,19)(17,20)(18,21)(23,28,29,26)(25,27)(30,31,35,41,37,40,50,58,36,42,49,59,34,51,33,39)(32,43,46,57,45,38,44,54,47,56,60,53,52,55,48,61);(14,16,18,24)(15,19,21,22)(17,23,25,29)(20,26,27,28)(30,32,36,47)(31,38,42,55)(33,48,50,44)(34,52,37,45)(35,46,49,60)(39,57,58,53)(40,56,51,43)(41,54,59,61);(14,17,18,25)(15,20,21,27)(16,23,24,29)(19,26,22,28)(30,33,34,49,36,50,37,35)(31,39,51,59,42,58,40,41)(32,44,52,46,47,48,45,60)(38,53,43,54,55,57,56,61)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 4486);