Finite Group DatabasebetaFinite Group Databasebeta| order | 12068 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C431 · Schur multiplier (H2) = C2 |
| direct product | C2×D7×C431 |
| unambiguous semidirect product | C7⋊(C22×C431) |
| unambiguous semidirect product | (C2×C7)⋊(C2×C431) |
| Fitting, socle | C2×C7×C431 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C431 |
| Hall | C7×C431 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C431 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C431 |
| Frattini | C1 | G |
| Fitting, socle | C2×C7×C431 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C431 | |
20 conjugacy classes of subgroups.
4310 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C2×AGL(1,𝔽7)×C431, D3×D7×C431, C2×C3×D7×C431, … |
| maximal normal subgroup of | C2×AGL(1,𝔽7)×C431, C2×C3×D7×C431 |
| maximal quotient of | D3×D7×C431, C2×C3×D7×C431, C2×D21×C431 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12068, 8);