Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of ΓL(1,𝔽8)⋊Q8 |
| automorphism | Aut(G) = D4×AGL(1,𝔽7) (order 336) · Inn(G) = C2×AGL(1,𝔽7) · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊AGL(1,𝔽7) |
| unambiguous semidirect product | D28⋊C3 |
| Fitting | C4×C7 ⋊ C2×C3 |
| Hall | D28 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C7 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C22×C3 |
| Fitting | C4×C7 | C2×C3 |
| hypercenter | C4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C4 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 28 |
A minimal faithful permutation representation — 2 generators on 11 points:
(1,2,7)(3,4,6)(8,9,10,11);(2,3)(4,6)(5,7)(9,11)Polycyclic presentation:
2^2,3,2,7|b^2=d,b^a=bd,e^a=e^6,e^c=e^419 conjugacy classes — view the full character table.
| maximal subgroup of | #1132⋊C3, D56⋊C3, #1141⋊C3, … |
| maximal normal subgroup of | #1132⋊C3, D56⋊C3, #1141⋊C3, … |
| maximal quotient of | #1132⋊C3, D56⋊C3, Dic28⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 9);