Finite Group DatabasebetaFinite Group Databasebeta| order | 686 |
| isoclinism | isoclinic class of (C7⋊D7)⋊C7 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C7 |
| unambiguous semidirect product | (C7⋊D7)⋊C7 |
| Fitting, Sylow | H(3,𝔽7) ⋊ C2 |
| derived | C72 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C72 | C2×C7 |
| Frattini, socle | C7 | C7×D7 |
| Fitting | H(3,𝔽7) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 | |
20 conjugacy classes of subgroups.
| transitive degree | μt(G) = 49 |
Polycyclic presentation:
2,7^3|c^a=c^6,d^a=d^6,c^b=cd38 conjugacy classes — view the full character table.
| maximal subgroup of | ((C7⋊D7)⋊C7)⋊C2, C2×(C7⋊D7)⋊C7, #18012418, … |
| maximal normal subgroup of | ((C7⋊D7)⋊C7)⋊C2, C2×(C7⋊D7)⋊C7, C7×(C7⋊D7)⋊C7, … |
| maximal quotient of | (C7⋊Dic7)⋊C7, C2×(C7⋊D7)⋊C7, (C7⋊D7)⋊C49, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(686, 5);