Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C26 |
| direct product | C2×C13×#416 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C13 | D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C23×C13 | C2×D4 |
| prime radical | C22×C13×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ C23×C4×C13 ◁ G | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 35 points:
(1,2)(23,24,25,26,27,28,29,30,31,32,33,34,35);(3,4,5,6)(7,8)(9,16)(10,14)(11,19)(12,15)(13,17)(18,21)(20,22);(7,9)(8,13)(10,17)(11,20)(12,18)(14,21)(15,22)(16,19);(7,10,12,11)(8,14,15,19)(9,17,18,20)(13,21,22,16)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1624);