Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#61384 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C13 |
| prime radical | C2×C13×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#533 ◁ G | |
1726 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)(14,15,18,17)(16,20,21,19)(22,23)(24,27)(25,33)(26,30)(28,32)(29,34)(31,36)(35,37);(14,16,18,21)(15,19,17,20)(22,24,27,23)(25,31,35,29)(26,30,28,32)(33,34,37,36);(22,25,27,35)(23,29,24,31)(26,34,28,36)(30,37,32,33);(14,17,18,15)(16,19,21,20)(22,26,27,28)(23,30,24,32)(25,34,35,36)(29,37,31,33);(23,24)(29,31)(30,32)(33,37)715 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 54372);