Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C24 |
| direct product | C13×#3177 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 53 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,19)(17,18)(20,21)(22,23,27,31)(24,28,33,41)(25,35,38,50)(26,39,36,30)(29,43,44,37)(32,45,47,53)(34,49,46,42)(40,51,52,48);(15,18)(19,21)(22,24,27,33)(23,28,31,41)(25,32,38,47)(26,34,36,46)(29,40,44,52)(30,42,39,49)(35,45,50,53)(37,48,43,51);(22,25,36,43,27,38,26,37)(23,29,39,50,31,44,30,35)(24,32,46,51,33,47,34,48)(28,40,49,53,41,52,42,45);(14,16,17,20)(15,19,18,21)(24,33)(28,41)(32,47)(34,46)(40,52)(42,49)(45,53)(48,51)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1909);