Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#33443 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C13 | C22×D4 |
| commutator | C4 | C23×C8×C13 |
| Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C2×C13 | C2×C8×D4 |
| prime radical | C2×C13×#135 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C13 ◁ C2×C13×M32 ◁ G | |
648 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,24,18,25,33,42,20,22,35,38,34,23,39,41)(16,26,28,32,29,37,43,31,30,40,44,45,27,21,36,17);(14,16)(15,21)(17,24)(18,29)(19,28)(20,30)(22,37)(23,40)(25,26)(27,34)(31,38)(32,42)(33,43)(35,44)(36,39)(41,45);(14,17,19,26,18,32,33,37,20,31,35,40,34,45,39,21)(15,16,24,28,25,29,42,43,22,30,38,44,23,27,41,36);(15,22)(17,31)(21,37)(23,25)(24,38)(26,40)(32,45)(41,42)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26431);