Finite Group DatabasebetaFinite Group Databasebeta| order | 126 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C3×AGL(1,𝔽7) (order 126) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = C2×#72224657 |
| homology | Abelianization (H1) = C2×C9 · Schur multiplier (H2) = C1 |
| direct product | C2×C7⋊C9 |
| unambiguous semidirect product | (C2×C7)⋊C9 |
| derived, Sylow | C7 ⋊ C2×C9 |
| Hall | C2×C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C2×C9 |
| Frattini | C3 | C2×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C2×C3×C7 | C3 |
| cosocle | C3×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 126 |
A minimal faithful permutation representation — 2 generators on 18 points:
(1,2)(4,5,6)(7,9,8)(10,14,18,12,17,13,15,11,16);(3,4,5,7,6,8,9)30 conjugacy classes — view the full character table.
| maximal subgroup of | Dic7⋊C9, C4×C7⋊C9, C2×D7⋊C9, … |
| maximal normal subgroup of | Dic7⋊C9, C4×C7⋊C9, C2×D7⋊C9, … |
| maximal quotient of | C4×C7⋊C9, C22×C7⋊C9, C2×C7⋊C27, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(126, 2);