Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#1979 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #121 |
| commutator | C2×C4 | C22×C4×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| prime radical | C23×C13 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ G | |
276 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 112 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)(15,17)(18,19,23,28,24,27,22,36)(20,25,31,40,30,41,37,39)(21,32,34,42,35,46,33,26)(29,43,45,38,44,48,47,49);(14,15,16,17)(18,20,22,31,24,30,23,37)(19,25,28,39,27,41,36,40)(21,29,33,45,35,44,34,47)(26,38,32,43,42,49,46,48);(18,21,23,34,24,35,22,33)(19,26,36,46,27,42,28,32)(20,29,37,47,30,44,31,45)(25,38,40,48,41,49,39,43)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 711);