Finite Group DatabasebetaFinite Group Databasebeta| order | 42 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C2×AGL(1,𝔽7) (order 84) · Inn(G) = D7 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = D3×#64423⋊C2 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
| direct product | C3×D7 |
| Fitting, socle, Hall | C3×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | D7 |
| commutator, cosocle, hypocenter | C7 | C2×C3 |
| Frattini | C1 | G |
| Fitting, socle | C3×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 21 |
A minimal faithful permutation representation — 2 generators on 10 points:
(1,2,3)(5,6)(7,8)(9,10);(4,5,7,9,10,8,6)15 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D7, C2×C3×D7, D7⋊C9, … |
| maximal normal subgroup of | D3×D7, C2×C3×D7, D7⋊C9, … |
| maximal quotient of | C3×Dic7, C2×C3×D7, C9×D7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(42, 4);