Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C13 · Schur multiplier (H2) = C2 |
| direct product | C13×#1340 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | #327 |
| commutator | C2×C8 | C2×C4×C13 |
| Frattini, cosocle | C2×M16 | C22×C13 |
| prime radical | C23×C13 | D8 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ C2×C13×M16 ◁ G | |
104 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)(14,15)(16,25,27,42)(17,30,32,41)(18,23)(19,35)(20,24)(21,26)(22,33,39,34)(28,38,29,45)(31,40)(36,43)(37,44);(14,16,20,27)(15,21,24,37)(17,31,32,43)(18,29,19,28)(22,38,39,45)(23,40,35,36)(25,41,42,30)(26,33,44,34)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 72);