Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#12668 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #106 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | C4×#106 |
| prime radical | C23×C13 | #135 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
308 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)(14,15,19,25,20,27,33,40,21,26,18,24,32,39,17,23)(16,28,29,38,30,41,34,37,31,44,45,43,36,42,35,22);(14,16)(15,22,26,37)(17,29)(18,34)(19,35)(20,36)(21,31)(23,28,40,44)(24,41,25,42)(27,43,39,38)(30,32)(33,45);(14,17,32,18,21,33,20,19)(15,23,39,24,26,40,27,25)(16,29,30,34,31,45,36,35)(22,28,38,41,37,44,43,42)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5656);