Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C13×#540 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
382 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)(16,19)(17,18)(20,21)(23,27)(24,30)(26,32)(28,36)(31,33)(34,37);(15,18)(19,21)(22,23)(24,28)(25,27)(26,29)(30,37)(31,34)(32,35)(33,36);(22,24,25,31)(23,28,27,34)(26,33,32,30)(29,36,35,37);(14,16,17,20)(15,19,18,21)(23,27)(28,34)(29,35)(36,37)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1858);