Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#13200 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | #127 |
| commutator | C22×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1955 |
| prime radical | C13×#525 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C22×C8×C13 ◁ G | |
438 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)(14,15,19,25,20,24,36,40,21,16,22,28,18,27,35,43)(17,29,33,44,34,26,41,42,31,30,37,45,32,23,38,39);(14,16,19,28,20,27,36,43,21,15,22,25,18,24,35,40)(17,30,37,44,34,23,38,42,31,29,33,45,32,26,41,39);(14,17,18,32,21,31,20,34)(15,23,24,29,16,26,27,30)(19,33,35,38,22,37,36,41)(25,39,40,44,28,42,43,45)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 6188);