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 |
| direct product | C13×#2179 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C2×D8 |
| commutator | C8 | C22×C4×C13 |
| Frattini, cosocle | C2×C8 | C23×C13 |
| socle | C2×C13 | C4×D8 |
| prime radical | C13×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C8×C13 ◁ C4×C8×C13 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,19,24,20,26,36,25)(16,27,29,42,31,44,30,38)(17,23,32,41,33,22,18,34)(21,37,39,45,40,35,43,28);(14,16)(15,21)(17,28)(18,35)(19,29)(20,31)(22,38)(23,42)(24,39)(25,43)(26,40)(27,34)(30,36)(32,37)(33,45)(41,44);(14,17,19,32,20,33,36,18)(15,22,24,34,26,23,25,41)(16,28,29,37,31,45,30,35)(21,38,39,27,40,42,43,44)416 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 911);