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) = C22×C7 · Schur multiplier (H2) = C2 |
| direct product | C2×D3×C7 |
| unambiguous semidirect product | C3⋊(C22×C7) |
| unambiguous semidirect product | (C2×C3)⋊(C2×C7) |
| Fitting, socle | C2×C3×C7 ⋊ C2 |
| derived, Sylow | C3 ⋊ C22×C7 |
| Hall | C3×C7 ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | D3 |
| commutator, cosocle, hypocenter | C3 | C22×C7 |
| Frattini | C1 | G |
| Fitting, socle | C2×C3×C7 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 ◁ C2×C7 | |
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)(4,5)(6,7,8,9,10,11,12);(1,2)(3,4,5)42 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic7, #4245, #4247, … |
| maximal normal subgroup of | D3×Dic7, #4245, #4247, … |
| maximal quotient of | C7×Dic6, C4×D3×C7, C7×D12, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 13);