Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D3×D7 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = D3×D7 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C2×D3×D7 |
| unambiguous semidirect product | (D3×C7)⋊C22 |
| unambiguous semidirect product | (C3×D7)⋊C22 |
| unambiguous semidirect product | D21⋊C22 |
| Fitting, socle | C2×C3×C7 ⋊ C22 |
| derived, Hall | C3×C7 ⋊ C23 |
| Sylow | C7 ⋊ C22×D3 |
| Sylow | C3 ⋊ C22×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D3×D7 |
| commutator, cosocle, hypocenter | C3×C7 | C23 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C22 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
64 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4,5)(7,8)(9,10)(11,12);(4,5)(6,7,9,11,12,10,8);(1,2)30 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×D7, D7×D12, D3×D28, … |
| maximal normal subgroup of | C4×D3×D7, D7×D12, D3×D28, … |
| maximal quotient of | D7×Dic6, #66669, #66670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 50);