Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3414 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C24×C13 |
| socle | C2×C13 | C22×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
364 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 364 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) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,25)(17,23)(18,30)(19,34)(20,22)(21,27)(24,32)(26,42)(28,41)(29,39)(31,37)(33,44)(35,40)(36,43)(38,45);(14,16)(15,21)(17,26)(18,31)(19,35)(20,28)(22,36)(23,37)(24,40)(25,30)(27,32)(29,43)(33,45)(34,38)(39,42)(41,44);(15,22)(21,36)(23,34)(24,39)(25,41)(30,44)(37,38)(40,42);(14,17,20,19)(15,23,22,34)(16,26,28,35)(18,29,33,32)(21,37,36,38)(24,30,39,44)(25,42,41,40)(27,31,43,45)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2146);