Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#12646 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #102 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | C4×#102 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ C2×C4×C8×C13 ◁ G | |
428 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)(14,15,18,25,21,27,20,37)(16,28,31,40,34,43,33,38)(17,23,35,42,36,26,19,24)(22,32,39,29,41,45,44,30);(14,16,19,32,20,33,36,30,21,34,35,45,18,31,17,29)(15,22,26,40,25,39,24,43,27,41,23,38,37,44,42,28);(14,17,18,35,21,36,20,19)(15,23,25,42,27,26,37,24)(16,29,31,45,34,30,33,32)(22,38,39,28,41,40,44,43)832 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5634);