Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C2×C4 |
| direct product | C13×#1634 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| prime radical | C13×#555 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
218 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(19,21)(22,23,27,30)(24,31,33,36)(25,29,26,34)(28,35,37,32);(16,20)(18,21)(22,24,27,33)(23,28,30,37)(25,32,26,35)(29,36,34,31);(14,15,17,19)(16,21,20,18)(22,25,27,26)(23,29,30,34)(24,32,33,35)(28,36,37,31)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 366);