Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C22×C3×D4 (order 96) · Inn(G) = C22 · Out(G) = C23×C3 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽7)×#453 |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C3×C7×D4 |
| unambiguous semidirect product | C4⋊(C2×C3×C7) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3×C7 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C3×C7 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C7 ◁ G | |
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:
(1,2,3)(4,5,6,7,8,9,10)(11,12,13,14);(12,14)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 40);