Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61639 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#525 ◁ G | |
2174 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,20)(17,19)(18,21)(22,23)(24,33)(25,32)(26,31)(27,30)(28,35)(29,34)(36,37);(14,16)(15,18)(17,21)(19,20)(22,24)(23,28)(25,36)(26,35)(27,34)(29,32)(30,37)(31,33);(22,24)(23,29)(25,34)(26,35)(27,36)(28,32)(30,33)(31,37);(16,21)(18,20);(15,19)(18,20)(22,25,26,27)(23,30,31,32)(24,34,35,36)(28,29,33,37)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54627);