Finite Group DatabasebetaFinite Group Databasebeta| order | 168 |
| isoclinism | isoclinic class of Dic6 |
| automorphism | Aut(G) = C2×C3×D3×D4 (order 288) · Inn(G) = C2×D3 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×D12 |
| unambiguous semidirect product | C4⋊(D3×C7) |
| Fitting | C3×C4×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ D4 |
| Sylow | C3 ⋊ C7×D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7 | C2×D3 |
| commutator, cosocle | C2×C3 | C22×C7 |
| Frattini | C2 | C2×D3×C7 |
| Fitting | C3×C4×C7 | C2 |
| socle | C2×C3×C7 | C22 |
| hypercenter | C4×C7 | D3 |
| hypocenter | C3 | C7×D4 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
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);(9,11)(13,14)63 conjugacy classes — view the full character table.
| maximal subgroup of | #66560, #66562, #66564, … |
| maximal normal subgroup of | #66560, #66562, #66564, … |
| maximal quotient of | C7×#204, C7×D24, C7×Dic12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 31);