Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C13 · Schur multiplier (H2) = C23 |
| direct product | C13×#1280 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C13 | C22⋊C4 |
| commutator | C22 | C4×C8×C13 |
| Frattini, cosocle | C23×C4 | C22×C13 |
| prime radical | C24×C13 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C13 ◁ C23×C4×C13 ◁ G | |
264 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(18,19)(20,24,25,22)(21,23)(26,27,28,30,29,31,32,33);(14,15,16,17)(18,20)(19,22)(21,25)(23,24)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 12);