Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C13 · Schur multiplier (H2) = C8 |
| direct product | C13×#1717 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C13 |
| prime radical | C13×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C42×C13 ◁ G | |
208 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 208 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,18)(16,20)(19,21)(23,28)(24,29)(25,26);(14,15,16,18,17,19,20,21)(22,23,26,24,27,29,25,28);(15,19)(18,21)(22,24,26,29,27,28,25,23)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 449);