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) = C1 |
| direct product | C7×Dic6 |
| unambiguous semidirect product | C3⋊(C7×Q8) |
| Hall | C3×C7 ⋊ Q8 |
| Sylow | C3 ⋊ C7×Q8 |
| 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, prime radical | C2×C3×C7 | C22 |
| hypercenter | C4×C7 | D3 |
| hypocenter | C3 | C7×Q8 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C7 ◁ C4×C7 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 168 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2,3,4,5,6,7)(8,9,10,11)(12,15,14,13)(16,17,18);(8,12,10,14)(9,13,11,15)(17,18)63 conjugacy classes — view the full character table.
| maximal subgroup of | #66563, #66568, #66569, … |
| maximal normal subgroup of | #66563, #66568, #66569, … |
| maximal quotient of | C7×Dic3⋊C4, C7×C4⋊Dic3, C2×C7×Dic6, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(168, 29);