Finite Group DatabasebetaFinite Group Databasebeta| order | 84 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = C22×C3×D3 (order 72) · Inn(G) = D3 · Out(G) = C22×C3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽7)×#3811 |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C1 |
| direct product | C7×Dic3 |
| unambiguous semidirect product | C3⋊(C4×C7) |
| derived, Sylow | C3 ⋊ C4×C7 |
| Hall | C3×C7 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D3 |
| commutator, hypocenter | C3 | C4×C7 |
| Frattini | C2 | D3×C7 |
| Fitting, socle, prime radical | C2×C3×C7 | C2 |
| cosocle | C2×C3 | C2×C7 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C7 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(2,3)(4,7,6,5)(8,9,10,11,12,13,14);(1,2,3)42 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×D7, D21⋊C4, #4246, … |
| maximal normal subgroup of | Dic3×D7, D21⋊C4, #4246, … |
| maximal quotient of | C7×C3⋊C8, C2×C7×Dic3, C7×Dic9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 3);