Finite Group DatabasebetaFinite Group Databasebeta| order | 686 |
| isoclinism | isoclinic class of D49⋊C7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | D49⋊C7 |
| unambiguous semidirect product | 7−1+2⋊C2 |
| Fitting, Sylow | 7−1+2 ⋊ C2 |
| derived | C49 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C49 | C2×C7 |
| Frattini, socle | C7 | C7×D7 |
| Fitting | 7−1+2 | C2 |
| LCS | G ▷ C49 | |
| UCS | C1 | |
15 conjugacy classes of subgroups.
| transitive degree | μt(G) = 49 |
Polycyclic presentation:
2,7^3|c^7=d,c^a=c^6d^6,d^a=d^6,c^b=cd38 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D49⋊C7, #14558877⋊C2, #18012424, … |
| maximal normal subgroup of | C2×D49⋊C7, C7×D49⋊C7, #72215772, … |
| maximal quotient of | Dic49⋊C7, C2×D49⋊C7, #18012411, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(686, 6);