Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C24×C3 (order 48) · Inn(G) = C1 · Out(G) = C24×C3 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,ℤ/8)×AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C168 |
| abelian | C3×C7×C8 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C4 | C2×C3×C7 |
| socle, prime radical | C2×C3×C7 | C4 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 1 generator on 18 points:
(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 6);