Finite Group DatabasebetaFinite Group Databasebeta| order | 10668 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #19128625⋊C2 |
| unambiguous semidirect product | #19128629⋊C2 |
| unambiguous semidirect product | #19128633⋊C2 |
| Fitting, socle, derived, Hall | C7×C127 ⋊ C22×C3 |
| Hall | D7×D127 ⋊ C3 |
| Hall | #14581887 ⋊ C22 |
| Sylow | C127 ⋊ C2×AGL(1,𝔽7) |
| Sylow | C7 ⋊ C2×D127⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C7×C127 | C22×C3 |
| LCS | G ▷ C7×C127 | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
Polycyclic presentation:
2^2,3,7,127|e^a=e^126,d^b=d^6,d^c=d^2,e^c=e^19119 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×D127⋊C3, C3×#19128625⋊C2, #19128625⋊D3, … |
| maximal normal subgroup of | AGL(1,𝔽7)×D127⋊C3, C3×#19128625⋊C2 |
| maximal quotient of | #30630664⋊C2, C3×#19128625⋊C2, #19128625⋊D3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10668, 41);