Finite Group DatabasebetaFinite Group Databasebeta| order | 686 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C72 · Schur multiplier (H2) = C72 |
| direct product | C2×H(3,𝔽7) |
| unambiguous semidirect product | (C2×C72)⋊C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C72 |
| commutator, Frattini, cosocle | C7 | C2×C72 |
| LCS | G ▷ C7 ▷ C1 | |
| UCS | C1 ◁ C2×C7 ◁ G | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 51 |
| transitive degree | μt(G) = 98 |
A minimal faithful permutation representation — 2 generators on 51 points:
(1,2)(3,4,6,10,17,28,37)(5,8,12,19,30,39,46)(7,13,23,35,44,51,45)(9,15,22,32,41,38,47)(11,20,33,43,50,36,14)(16,26,21,34,29,40,48)(18,31,42,49,27,25,24);(4,7,14,25,26,15,8)(6,11,21,12,13,24,22)(10,18,19,20,32,23,34)(17,29,35,41,33,30,31)(28,38,42,44,39,40,43)(37,46,47,48,49,50,51)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(686, 10);