Finite Group DatabasebetaFinite Group Databasebeta| order | 84 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C3×D3 (order 72) · Inn(G) = C1 · Out(G) = C22×C3×D3 |
| holomorph | G ⋊ Aut(G) = D3×S4×AGL(1,𝔽7) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C3×C7 |
| multiplicative | (ℤ/129)ˣ, (ℤ/147)ˣ, (ℤ/172)ˣ, (ℤ/196)ˣ, (ℤ/258)ˣ, (ℤ/294)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(5,6,7)(8,9,10,11,12,13,14);(3,4)84 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(84, 15);