Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C13×#1954 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | C23 |
| commutator | C22 | C22×C8×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C23×C13 | C22×C4 |
| prime radical | C22×D4×C13 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ G | |
396 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 396 conjugacy classes of subgroups — too many to draw. Only its 160 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)(15,17)(19,21)(22,23,25,28,26,24,27,29);(14,15,16,17)(22,24,25,29,26,23,27,28);(18,19,20,21)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 686);