Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C13 · Schur multiplier (H2) = C23 |
| direct product | C2×C13×Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | D4 |
| commutator | C4 | C22×C8×C13 |
| Frattini, cosocle | C42 | C23×C13 |
| socle, prime radical | C23×C13 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ C2×C42×C13 ◁ G | |
280 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 31 points:
(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31);(4,7)(5,9)(8,10)(11,12,13,15,14,16,17,18);(3,4,6,8)(5,10,9,7)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 207);