Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C2×C3×D3×D4 (order 288) · Inn(G) = D3 · Out(G) = C2×C3×D4 |
| homology | Abelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×C7×Dic3 |
| unambiguous semidirect product | (C2×C3)⋊(C4×C7) |
| derived, Sylow | C3 ⋊ C2×C4×C7 |
| Hall | C3×C7 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C7 | D3 |
| commutator, hypocenter | C3 | C2×C4×C7 |
| Frattini | C2 | C2×D3×C7 |
| Fitting, socle, prime radical | C22×C3×C7 | C2 |
| cosocle | C2×C3 | C22×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C22×C7 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(4,5)(6,7,8,9)(10,11,12,13,14,15,16);(1,2)(3,4,5)84 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×Dic7, (C2×D7)⋊Dic3, #66575, … |
| maximal normal subgroup of | Dic3×Dic7, (C2×D7)⋊Dic3, #66575, … |
| maximal quotient of | C2×C7×C3⋊C8, C7×#208, C4×C7×Dic3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 32);