Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×D28 |
| unambiguous semidirect product | C4⋊(C3×D7) |
| Fitting | C3×C4×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ D4 |
| Sylow | C7 ⋊ C3×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C2×D7 |
| commutator, cosocle | C2×C7 | C22×C3 |
| Frattini | C2 | C2×C3×D7 |
| Fitting | C3×C4×C7 | C2 |
| socle | C2×C3×C7 | C22 |
| hypercenter | C3×C4 | D7 |
| hypocenter | C7 | C3×D4 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2×C3 ◁ C3×C4 | |
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);(5,7)(9,14)(10,13)(11,12)51 conjugacy classes — view the full character table.
| maximal subgroup of | #66560, #66561, #66563, … |
| maximal normal subgroup of | #66560, #66561, #66563, … |
| maximal quotient of | C3×#1132, C3×D56, C3×Dic28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 26);