Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C22 |
| direct product | C13×#1278 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C13 | D4 |
| commutator | C4 | C4×C8×C13 |
| Frattini, cosocle | C2×C42 | C22×C13 |
| socle | C22×C13 | #95 |
| prime radical | C23×C13 | C4⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C13 ◁ C2×C42×C13 ◁ G | |
172 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 2 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15)(16,18)(17,20)(19,21)(22,23,25,28,27,29,26,24);(15,18,20,19)(22,24,26,29,27,28,25,23)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 10);