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) = C24×C4 |
| direct product | C2×C13×#427 |
| 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 | C24×C13 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C13 ◁ C23×C4×C13 ◁ G | |
512 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)(11,20)(12,15)(13,18)(17,21)(19,22);(7,9,11,18,12,19,10,17)(8,13,14,16,15,21,20,22);(7,10,12,11)(8,14,15,20)(9,17,19,18)(13,16,21,22)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1641);