Finite Group DatabasebetaFinite Group Databasebeta| order | 126 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×D3 (order 72) · Inn(G) = D3 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = D33×AGL(1,𝔽7) |
| homology | Abelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C1 |
| direct product | C3×D3×C7 |
| unambiguous semidirect product | C3⋊(C2×C3×C7) |
| Fitting, socle, Hall | C32×C7 ⋊ C2 |
| derived | C3 ⋊ C2×C3×C7 |
| Sylow | C32 ⋊ C2×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | D3 |
| commutator, cosocle, hypocenter | C3 | C2×C3×C7 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C3×C7 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,2,3)(5,6)(7,8,9,10,11,12,13);(4,5,6)63 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D7, D32×C7, D3×D21, … |
| maximal normal subgroup of | C3×D3×D7, D32×C7, D3×D21, … |
| maximal quotient of | C3×C7×Dic3, C2×C3×D3×C7, C3×C7×D9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(126, 12);