Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #459 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1802 |
| 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 | |
400 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)(14,15,20,24,21,26,18,34)(16,25,19,35,29,42,28,22)(17,30,32,45,33,37,31,41)(23,36,38,44,40,43,39,27);(14,16,18,28,21,29,20,19)(15,22,24,25,26,35,34,42)(17,27,31,43,33,44,32,36)(23,37,38,41,40,30,39,45);(14,17)(15,23)(16,27)(18,31)(19,36)(20,32)(21,33)(22,37)(24,38)(25,41)(26,40)(28,43)(29,44)(30,35)(34,39)(42,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5810);