Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#33656 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | #481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C13 |
| socle | C2×C13 | C2×#481 |
| prime radical | C13×#3054 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C13×M16 ◁ G | |
556 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)(16,25)(17,22)(18,30)(19,34)(20,24)(21,28)(23,33)(26,40)(27,41)(29,39)(31,45)(32,38)(35,42)(36,43)(37,44);(18,31)(23,39)(25,41)(28,44)(29,33)(30,45)(32,43)(40,42);(14,16,19,35,20,27,17,26)(15,21,22,36,24,37,34,38)(18,32,33,44,31,43,29,28)(23,40,30,41,39,42,45,25);(14,17,20,19)(15,22,24,34)(16,26,27,35)(18,29,31,33)(21,36,37,38)(23,30,39,45)(25,40,41,42)(28,43,44,32)520 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 26644);