Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C23×C3×D3 (order 144) · Inn(G) = D3 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C7×C8 · Schur multiplier (H2) = C1 |
| direct product | C7×C3⋊C8 |
| unambiguous semidirect product | C3⋊(C7×C8) |
| derived, Sylow | C3 ⋊ C7×C8 |
| Hall | C3×C7 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4×C7 | D3 |
| commutator, hypocenter | C3 | C7×C8 |
| Frattini | C4 | D3×C7 |
| Fitting | C3×C4×C7 | C2 |
| socle, prime radical | C2×C3×C7 | C4 |
| cosocle | C3×C4 | C2×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C4×C7 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,11,10,9,7,8,6,5)(12,13,14,15,16,17,18);(1,2,3)84 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C8, D21⋊C8, #66557, … |
| maximal normal subgroup of | D7×C3⋊C8, D21⋊C8, #66557, … |
| maximal quotient of | C7×C3⋊C16, C2×C7×C3⋊C8, C7×C9⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 3);