Finite Group DatabasebetaFinite Group Databasebeta| order | 84 |
| isoclinism | isoclinic class of D7 |
| automorphism | Aut(G) = C22×AGL(1,𝔽7) (order 168) · Inn(G) = D7 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = D3×#72210376 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | C2×C3×D7 |
| Fitting, socle | C2×C3×C7 ⋊ C2 |
| derived, Sylow | C7 ⋊ C22×C3 |
| Hall | C3×C7 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D7 |
| commutator, cosocle, hypocenter | C7 | C22×C3 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C2 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
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)(7,8)(9,10)(11,12);(1,2)(6,7,9,11,12,10,8)30 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D7, #4245, #4246, … |
| maximal normal subgroup of | Dic3×D7, #4245, #4246, … |
| maximal quotient of | C3×Dic14, C3×C4×D7, C3×D28, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 12);