Finite Group DatabasebetaFinite Group Databasebeta| order | 3328 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C13 · Schur multiplier (H2) = C28 |
| direct product | C13×#62150 |
| 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 ◁ C4×C13×#41 ◁ G | |
1996 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)(15,18)(16,19)(20,21)(22,23,26,29)(24,30,31,27)(25,33,34,28)(32,35,37,36);(14,15,17,18)(16,19,20,21)(22,24)(23,27)(25,32)(26,31)(28,35)(29,30)(33,36)(34,37);(14,16,17,20)(15,19,18,21);(14,15,17,18)(16,19,20,21)(24,31)(27,30)(32,37)(35,36);(14,15,17,18)(16,19,20,21)(22,25,26,34)(23,28,29,33)(24,32,31,37)(27,35,30,36)676 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3328, 55138);