Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #357 |
| commutator | C16 | C2×C4×C13 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| prime radical | C13×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C8×C13 ◁ C2×C13×C16 ◁ G | |
74 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 2 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(15,20,21,36)(16,24)(17,29)(18,31)(22,40,39,44)(23,41,33,38)(25,30,37,35)(26,32)(27,45)(28,34)(42,43);(14,15)(16,25)(17,30)(18,33)(19,21)(20,34)(22,29)(23,31)(24,36)(26,39)(27,40)(28,37)(32,35)(38,43)(41,45)(42,44)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 150);