Finite Group DatabasebetaFinite Group Databasebeta| order | 686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | H(3,𝔽7) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C7 | C7⋊D7 |
| commutator, Fitting, cosocle, hypocenter | H(3,𝔽7) | C2 |
| LCS | G ▷ H(3,𝔽7) | |
| UCS | C1 ◁ C7 | |
38 conjugacy classes of subgroups.
| transitive degree | μt(G) = 49 |
Polycyclic presentation:
2,7^3|b^a=b^6,c^a=c^6,c^b=cd38 conjugacy classes — view the full character table.
| maximal subgroup of | #13063133, ((C7⋊D7)⋊C7)⋊C2, C2×#10621859, … |
| maximal normal subgroup of | #13063133, ((C7⋊D7)⋊C7)⋊C2, C2×#10621859, … |
| maximal quotient of | #13063129, C2×#10621859, #18012412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(686, 8);