Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#29069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C13 |
| prime radical | C2×C13×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C4×C13×C22⋊C4 ◁ G | |
742 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1664 |
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)(16,20)(17,18)(19,21)(22,23)(24,28)(25,27)(26,29)(30,31)(32,36)(33,37)(34,38)(35,41)(39,43)(40,45)(42,44);(15,18)(19,20)(22,24,25,29)(23,26,27,28)(30,32,39,40,34,41,33,42)(31,35,43,44,38,36,37,45);(23,27)(26,28)(30,32,34,41)(31,36,38,35)(33,40,39,42)(37,45,43,44);(14,16,17,21)(15,19,18,20)(23,27)(26,28)(30,33,34,39)(31,37,38,43)(32,40,41,42)(35,44,36,45)598 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 22057);