Finite Group DatabasebetaFinite Group Databasebeta| order | 126 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×C3×AGL(1,𝔽7) (order 252) · Inn(G) = D7 · Out(G) = C2×C32 |
| holomorph | G ⋊ Aut(G) = AGL(1,ℤ/9)×#64423⋊C2 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C9×D7 |
| Fitting, Hall | C7×C9 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | D7 |
| commutator, hypocenter | C7 | C2×C9 |
| Frattini | C3 | C3×D7 |
| Fitting | C7×C9 | C2 |
| socle, cosocle | C3×C7 | C2×C3 |
| prime radical | C3×D7 | C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C9 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,3)(4,5)(6,7)(8,9,11,10,12,14,13,15,16);(1,2,4,6,7,5,3)45 conjugacy classes — view the full character table.
| maximal subgroup of | D7×D9, C2×C9×D7, D7⋊C27, … |
| maximal normal subgroup of | D7×D9, C2×C9×D7, D7⋊C27, … |
| maximal quotient of | C9×Dic7, C2×C9×D7, D7×C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(126, 4);