Finite Group DatabasebetaFinite Group Databasebeta| order | 7686 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊#14579421 |
| Fitting, socle, derived | C3×C7×C61 ⋊ C2×C3 |
| Hall | C3×#13046196 ⋊ C2 |
| Hall | C7×C61 ⋊ C3×D3 |
| Sylow | C61 ⋊ D3×ΓL(1,𝔽8) |
| Sylow | C7 ⋊ C3⋊(D61⋊C3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C3×C7×C61 | C2×C3 |
| LCS | G ▷ C3×C7×C61 | |
| UCS | C1 | |
36 conjugacy classes of subgroups.
Polycyclic presentation:
2,3^2,7,61|c^a=c^2,e^a=e^60,d^b=d^2,e^b=e^13225 conjugacy classes — view the full character table.
| maximal subgroup of | #14579425⋊D3, #14579429⋊D3, D3×#14579421, … |
| maximal normal subgroup of | #14579425⋊D3, #14579429⋊D3, D3×#14579421, … |
| maximal quotient of | C3⋊#19103968, C2×C3⋊#14579421, C5×C3⋊#14579421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(7686, 51);