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) = C3×C4 · Schur multiplier (H2) = C1 |
| direct product | C3×Dic7 |
| derived, Sylow | C7 ⋊ C3×C4 |
| Hall | C3×C7 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C3 | D7 |
| commutator, hypocenter | C7 | C3×C4 |
| Frattini | C2 | C3×D7 |
| Fitting, socle, prime radical | C2×C3×C7 | C2 |
| cosocle | C2×C7 | C2×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C2×C3 | |
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)(4,5)(6,7)(8,11,10,9)(12,13,14);(1,2,4,6,7,5,3)30 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic7, D21⋊C4, #4247, … |
| maximal normal subgroup of | D3×Dic7, D21⋊C4, #4247, … |
| maximal quotient of | C3×C7⋊C8, C2×C3×Dic7, C9×Dic7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 4);