Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#12998 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #447 |
| commutator | C2×C8 | C22×C4×C13 |
| Frattini, cosocle | C22×C8 | C23×C13 |
| socle | C2×C13 | #1874 |
| prime radical | C13×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C13×M16 ◁ G | |
500 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)(14,15,18,24,19,25,32,39,20,22,34,40,33,41,17,23)(16,26,29,36,30,42,27,43,28,44,31,21,35,37,45,38);(14,16,19,35,20,28,33,30)(15,21,25,43,22,36,41,38)(17,31,18,27,32,29,34,45)(23,42,24,26,39,37,40,44);(15,22)(17,32)(18,34)(21,36)(25,41)(27,45)(29,31)(38,43)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 5986);