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×#3127 |
| 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 | |
358 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)(22,23,26,30)(24,32,34,28)(25,36,37,45)(27,40,39,31)(29,41,44,38)(33,48,49,53)(35,43,51,47)(42,50,46,52);(15,18)(19,21)(22,24)(23,28)(25,33)(26,34)(27,35)(29,42)(30,32)(31,43)(36,53)(37,49)(38,50)(39,51)(40,47)(41,52)(44,46)(45,48);(22,25,26,37)(23,29,30,44)(24,33,34,49)(27,41,39,38)(28,42,32,46)(31,45,40,36)(35,52,51,50)(43,48,47,53);(14,16,17,20)(15,19,18,21)(24,34)(28,32)(33,49)(35,51)(42,46)(43,47)(48,53)(50,52)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1859);