Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C13 · Schur multiplier (H2) = C22×C4 |
| direct product | C13×#2140 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C2×D8 |
| commutator | C8 | C22×C4×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| prime radical | C22×C13×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C22×C4×C13 ◁ C22×C8×C13 ◁ G | |
260 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 260 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) = 416 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17)(18,19)(20,25)(21,24)(22,31)(23,27)(26,29)(28,32)(30,33);(19,24)(20,22)(21,29)(25,26)(27,33)(28,30)(31,32);(18,20,23,22)(19,25,27,31)(21,28,30,29)(24,32,33,26)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 872);