Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C13 · Schur multiplier (H2) = C210 |
| direct product | C13×#34257 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C25 |
| commutator | C22 | C24×C4×C13 |
| Frattini, cosocle | C23 | C25×C13 |
| prime radical | C22×C13×2−1+4 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ G | |
5050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,21,20)(23,24)(26,34)(28,30)(32,37);(22,23,27,24)(25,30,33,28)(26,29,34,31)(32,36,37,35);(14,15,16,17)(18,20,21,19)(22,24,27,23)(25,30,33,28)(26,31,34,29)(32,36,37,35);(22,25,27,33)(23,28,24,30)(26,32,34,37)(29,35,31,36);(22,26)(23,29)(24,31)(25,32)(27,34)(28,35)(30,36)(33,37)1144 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 27245);