Finite Group DatabasebetaFinite Group Databasebeta| order | 126 |
| isoclinism | isoclinic class of D21 |
| automorphism | Aut(G) = C2×D3×AGL(1,𝔽7) (order 504) · Inn(G) = D21 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = D3×#72213636 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D21 |
| unambiguous semidirect product | C3⋊(C3×D7) |
| Fitting, socle, Hall | C32×C7 ⋊ C2 |
| derived | C3×C7 ⋊ C2×C3 |
| Sylow | C32 ⋊ D7 |
| Sylow | C7 ⋊ C3×D3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D21 |
| commutator, cosocle, hypocenter | C3×C7 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C32×C7 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C3 | |
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)(8,9)(10,11)(12,13);(4,5,6)(7,8,10,12,13,11,9)36 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D3×D7, D3×D21, (C3⋊D3)⋊D7, … |
| maximal normal subgroup of | C3×D3×D7, D3×D21, (C3⋊D3)⋊D7, … |
| maximal quotient of | C3×Dic21, C2×C3×D21, C3×D63, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(126, 13);