Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C13 · Schur multiplier (H2) = C23×C4 |
| direct product | C8×C13×C22⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C13 | C22 |
| commutator | C2 | C2×C4×C8×C13 |
| Frattini, cosocle | C22×C4 | C23×C13 |
| socle | C23×C13 | C22×C4 |
| prime radical | C24×C13 | C2×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C13 ◁ G | |
384 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 216 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)(17,18,19,20,21,22,23,24,25,26,27,28,29);(9,10,12,14)(11,13,15,16);(10,13)(14,16)1040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 483);