Finite Group DatabasebetaFinite Group Databasebeta| order | 84 |
| 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) = #72222273 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D42 |
| direct product | C2×D21 |
| unambiguous semidirect product | (C2×C3)⋊D7 |
| unambiguous semidirect product | (C2×C7)⋊D3 |
| Fitting, socle | C2×C3×C7 ⋊ C2 |
| derived, Hall | C3×C7 ⋊ C22 |
| Sylow | C7 ⋊ C2×D3 |
| Sylow | C3 ⋊ C2×D7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D21 |
| commutator, cosocle, hypocenter | C3×C7 | C22 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C2 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 42 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2)(3,4,5)(6,7,8,9,10,11,12);(4,5)(7,12)(8,11)(9,10)24 conjugacy classes — view the full character table.
| maximal subgroup of | D21⋊C4, #4246, #4247, … |
| maximal normal subgroup of | D21⋊C4, #4246, #4247, … |
| maximal quotient of | Dic42, C4×D21, D84, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 14);