Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×SD128 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | D32 |
| commutator, Frattini, cosocle | C32 | C22×C13 |
| prime radical | C13×D32 | C2 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C4×C13 ◁ C8×C13 ◁ C13×C16 ◁ C13×C32 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77);(15,45)(16,76)(17,43)(18,74)(19,41)(20,72)(21,39)(22,70)(23,37)(24,68)(25,35)(26,66)(27,33)(28,64)(29,31)(30,62)(32,60)(34,58)(36,56)(38,54)(40,52)(42,50)(44,48)(47,77)(49,75)(51,73)(53,71)(55,69)(57,67)(59,65)(61,63)455 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 162);