Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C1 |
| direct product | C13×#1426 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #367 |
| commutator | C16 | C2×C4×C13 |
| Frattini, cosocle | C2×C16 | C22×C13 |
| prime radical | C22×C13 | SD32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C8×C13 ◁ C2×C13×C16 ◁ G | |
52 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)(15,20,22,33)(16,25)(17,29)(18,30)(21,36)(23,37,40,45)(24,43,41,35)(26,31)(27,39)(28,38)(32,44,42,34);(14,15,21,37,17,23,16,20,30,41,25,34,26,32,36,43,19,22,39,45,31,40,28,33,18,24,38,44,29,42,27,35)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 158);