Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C4 |
| direct product | C13×#2245 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C13 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C13 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ C4×C8×C13 ◁ G | |
132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 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)(22,23,29,35)(24,31,39,50)(25,30,41,48)(26,33,43,45)(27,32,36,51)(28,44,42,34)(37,49,52,47)(38,53,40,46);(15,18)(19,21)(22,24,36,52,28,38,26,25,29,39,27,37,42,40,43,41)(23,30,45,53,34,47,32,31,35,48,33,46,44,49,51,50);(14,16,17,20)(15,19,18,21)(22,25,26,40,42,52,36,39,29,41,43,38,28,37,27,24)(23,31,32,49,44,53,45,48,35,50,51,47,34,46,33,30)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 977);