Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#2141 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×D8 |
| commutator | C8 | C22×C4×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| prime radical | C23×C13 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
220 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,29)(21,26)(22,35)(23,39)(24,28)(25,31)(27,37)(30,43)(32,49)(33,46)(34,44)(36,42)(38,45)(40,47)(41,48);(18,20,24,33)(19,25,28,41)(21,32,23,30)(22,36,38,47)(26,42,39,40)(27,43,44,49)(29,45,46,35)(31,34,48,37);(18,21,24,23)(19,26,28,39)(20,30,33,32)(22,34,38,37)(25,40,41,42)(27,35,44,45)(29,43,46,49)(31,47,48,36)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 873);