Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3162 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C13 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C22×C13 | C22×D4 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C13 ◁ C2×C42×C13 ◁ G | |
348 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)(14,15,17,19)(16,20,21,18)(22,23)(24,31)(25,29)(26,35)(27,30)(28,34)(32,37)(33,36);(14,16,17,21)(15,18,19,20)(24,32)(28,36)(31,37)(33,34);(22,24,26,34,27,32,25,33)(23,28,29,31,30,36,35,37);(22,25,27,26)(23,29,30,35)(24,33,32,34)(28,31,36,37)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1894);