Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#13117 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #476 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1920 |
| prime radical | C23×C13 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C4×C8×C13 ◁ G | |
254 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)(14,15,17,21,20,25,19,23)(16,26,29,40,30,43,27,42)(18,32,31,41,35,39,34,24)(22,33,37,28,38,44,36,45);(15,21,25,23)(22,36,38,37)(24,32,41,39)(26,42,43,40);(14,16,17,27,20,30,19,29)(15,22,23,37,25,38,21,36)(18,33,31,45,35,44,34,28)(24,40,39,43,41,42,32,26)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6105);