Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C13 · Schur multiplier (H2) = C24×C4 |
| direct product | C13×#1436 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C13 | C23 |
| commutator | C22 | C2×C42×C13 |
| Frattini, cosocle | C24 | C23×C13 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C13 ◁ G | |
612 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1664 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,17)(18,19,21,20)(23,25)(26,27,29,28);(22,23)(24,25)(26,27,29,28);(14,15)(16,17)(18,20,21,19)(22,23)(24,25)(26,28,29,27)728 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 168);