Finite Group DatabasebetaFinite Group Databasebeta| order | 1664 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C13 · Schur multiplier (H2) = C25 |
| direct product | C13×#2957 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C13 | C22×D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| socle | C2×C13 | C2×C4×D4 |
| prime radical | C2×C13×#135 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C13 ◁ C2×C13×M16 ◁ G | |
452 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)(14,15,19,24,20,23,18,32)(16,25,27,21,28,36,33,35)(17,22,31,37,29,38,30,42)(26,39,41,34,40,44,43,45);(14,16)(15,21)(17,26)(18,33)(19,27)(20,28)(22,34)(23,35)(24,36)(25,32)(29,40)(30,43)(31,41)(37,44)(38,45)(39,42);(14,17,19,31,20,29,18,30)(15,22,24,37,23,38,32,42)(16,26,27,41,28,40,33,43)(21,34,36,44,35,45,25,39);(15,23)(17,29)(21,35)(24,32)(25,36)(26,40)(30,31)(41,43)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1689);