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×#12651 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #442 |
| commutator | C8 | C2×C42×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1762 |
| prime radical | C2×C13×#135 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C2×C13×#97 ◁ G | |
412 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)(14,15,18,25,21,26,20,36)(16,27,30,42,31,43,34,38)(17,23,19,35,33,41,32,24)(22,28,37,29,40,44,39,45);(14,16,19,28,18,34,33,45,21,31,32,44,20,30,17,29)(15,22,23,38,36,37,24,27,26,40,41,42,25,39,35,43);(14,17,18,19,21,33,20,32)(15,23,25,35,26,41,36,24)(16,28,30,29,31,44,34,45)(22,27,37,42,40,43,39,38)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5639);