Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #3415 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3417 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | C22×D8 |
| commutator, Frattini, cosocle | C8 | C24×C13 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C13×#41 ◁ C13×#131 ◁ G | |
350 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 350 conjugacy classes of subgroups — too many to draw. Only its 180 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 77 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 77 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,20,25)(16,26,30,54)(17,22,31,45)(18,34,38,52)(19,39,36,24)(21,41,44,28)(23,48,50,37)(27,53,55,75)(29,58,40,49)(32,61,59,74)(33,64,60,47)(35,51,66,43)(42,67,68,56)(46,71,70,63)(57,76,65,72)(62,73,77,69);(14,16,20,30)(15,21,25,44)(17,27,31,55)(18,35,38,66)(19,40,36,29)(22,42,45,68)(23,49,50,58)(24,51,39,43)(26,52,54,34)(28,48,41,37)(32,62,59,77)(33,65,60,57)(46,72,70,76)(47,73,64,69)(53,74,75,61)(56,71,67,63);(14,17)(15,22)(16,27)(18,32)(19,33)(20,31)(21,42)(23,46)(24,47)(25,45)(26,53)(28,56)(29,57)(30,55)(34,61)(35,62)(36,60)(37,63)(38,59)(39,64)(40,65)(41,67)(43,69)(44,68)(48,71)(49,72)(50,70)(51,73)(52,74)(54,75)(58,76)(66,77);(17,31)(22,45)(27,55)(32,59)(33,60)(42,68)(46,70)(47,64)(53,75)(56,67)(57,65)(61,74)(62,77)(63,71)(69,73)(72,76)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2149);