Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×D4 (order 96) · Inn(G) = C1 · Out(G) = C22×C3×D4 |
| holomorph | G ⋊ Aut(G) = D3×AGL(1,𝔽7)×B(4,𝔽2) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C2×C3×C4×C7 |
| multiplicative | (ℤ/203)ˣ, (ℤ/215)ˣ, (ℤ/245)ˣ, (ℤ/261)ˣ, (ℤ/406)ˣ, (ℤ/430)ˣ, (ℤ/490)ˣ, (ℤ/522)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C22×C3×C7 |
| socle, prime radical | C22×C3×C7 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
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:
(1,2,3,4)(7,8,9)(10,11,12,13,14,15,16);(5,6)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 39);