Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#3225 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C13×#449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 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)(23,28)(24,31)(25,27)(29,36)(32,37)(33,35);(14,16,17,21)(15,18,19,20)(22,23)(24,32)(25,34)(26,28)(27,30)(29,33)(31,36)(35,37);(16,21)(18,20)(22,24)(23,29)(25,31)(26,33)(27,35)(28,32)(30,36)(34,37);(15,19)(18,20)(22,25,26,27)(23,30,28,34)(24,31,33,35)(29,36,32,37)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1957);