Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of Dic42 |
| automorphism | Aut(G) = C22×D3×AGL(1,𝔽7) (order 1008) · Inn(G) = C2×D21 · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C3×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ D4 |
| Sylow | C7 ⋊ #67 |
| Sylow | C3 ⋊ C22⋊D7 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D21 |
| commutator, socle, cosocle | C2×C3×C7 | C22 |
| Fitting | C22×C3×C7 | C2 |
| hypercenter | C22 | D21 |
| hypocenter | C3×C7 | D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
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:
(1,2,4,6,7,5,3)(8,9,10)(12,14);(2,3)(4,5)(6,7)(9,10)(11,12)(13,14)Polycyclic presentation:
2^3,3,7|b^a=bc,d^a=d^2,e^a=e^645 conjugacy classes — view the full character table.
| maximal subgroup of | #66683, #66686, D7×#67, … |
| maximal normal subgroup of | #66683, #66686, D7×#67, … |
| maximal quotient of | #66629, #66631, #66632, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 38);