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×#2950 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C13 | C22×D4 |
| commutator | C4 | C23×C4×C13 |
| Frattini, cosocle | C2×C4 | C24×C13 |
| prime radical | C23×C13 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C13 ◁ C13×#117 ◁ G | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 832 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,17,16)(18,19)(20,28)(21,25)(22,34)(23,26)(24,30)(27,36)(29,42)(31,41)(32,40)(33,46)(35,38)(37,44)(39,45)(43,49)(47,48);(18,20,23,31)(19,24,26,38)(21,29,32,43)(22,35,36,30)(25,37,40,48)(27,41,34,28)(33,47,45,44)(39,49,46,42);(14,16,17,15)(18,21,23,32)(19,25,26,40)(20,29,31,43)(22,33,36,45)(24,37,38,48)(27,39,34,46)(28,42,41,49)(30,44,35,47);(19,26)(21,32)(24,38)(27,34)(28,41)(29,43)(33,45)(44,47)572 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1664, 1682);