Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#3289 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C13 | D42 |
| commutator, Frattini, cosocle | C2×C4 | C24×C13 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C13 ◁ C2×C4×C13 ◁ G | |
448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,20,26)(16,27,31,42)(17,23,33,41)(18,25,19,24)(21,34,38,30)(22,35,39,32)(28,36,45,37)(29,43,44,40);(14,16,20,31)(15,21,26,38)(17,28,33,45)(18,34,19,30)(22,36,39,37)(23,40,41,43)(24,42,25,27)(29,35,44,32);(14,17)(15,22)(16,28)(18,32)(19,35)(20,33)(21,36)(23,24)(25,41)(26,39)(27,43)(29,34)(30,44)(31,45)(37,38)(40,42);(14,17)(15,23)(16,29)(18,35)(19,32)(20,33)(21,37)(22,24)(25,39)(26,41)(27,43)(28,34)(30,45)(31,44)(36,38)(40,42)338 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 2021);