Finite Group DatabasebetaFinite Group Databasebeta| order | 12684 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #20263623⋊C2 |
| unambiguous semidirect product | #20263627⋊C2 |
| unambiguous semidirect product | #20263631⋊C2 |
| Fitting, socle, derived, Hall | C7×C151 ⋊ C22×C3 |
| Hall | D7×D151 ⋊ C3 |
| Hall | #15691748 ⋊ C22 |
| Sylow | C151 ⋊ C2×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×D151⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C151 | C22×C3 |
| LCS | G ▷ C7×C151 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7,151|e^a=e^150,d^b=d^6,d^c=d^2,e^c=e^32139 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D151⋊C3, C3×#20263623⋊C2, #20263623⋊D3, … |
| maximal normal subgroup of | AGL(1,𝔽7)×D151⋊C3, C3×#20263623⋊C2 |
| maximal quotient of | #34039752⋊C2, C3×#20263623⋊C2, #20263623⋊D3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12684, 26);