Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12753 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#12758 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #1790 |
| commutator | C2×D4 | C22×C4×C13 |
| Frattini, cosocle | C2×#41 | C23×C13 |
| prime radical | C13×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×#41 ◁ G | |
758 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)(14,15,19,24,20,25,17,22)(16,26,30,41,29,42,27,43)(18,32,31,39,33,23,34,38)(21,35,36,28,37,44,40,45);(14,16,20,29)(15,21,25,37)(17,27,19,30)(18,28,33,45)(22,36,24,40)(23,26,32,42)(31,44,34,35)(38,43,39,41);(14,17,20,19)(15,22,25,24)(16,27,29,30)(18,31,33,34)(21,36,37,40)(23,39,32,38)(26,41,42,43)(28,44,45,35)442 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5746);