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) = C2×C4 · Schur multiplier (H2) = C2 |
| direct product | Dic3×D7 |
| unambiguous semidirect product | (C3×D7)⋊C4 |
| derived, Hall | C3×C7 ⋊ C2×C4 |
| Sylow | C7 ⋊ C2×Dic3 |
| Sylow | C3 ⋊ C4×D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×D7 |
| commutator, hypocenter | C3×C7 | C2×C4 |
| Fitting, socle, cosocle | C2×C3×C7 | C22 |
| prime radical | C2×C3×D7 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,5)(6,7)(8,9,10);(1,2,4,6,7,5,3)(9,10)(11,12,13,14)30 conjugacy classes — view the full character table.
| maximal subgroup of | D7×Dic6, #66669, #66670, … |
| maximal normal subgroup of | D7×Dic6, #66669, #66670, … |
| maximal quotient of | D7×C3⋊C8, #66557, Dic3×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 12);