Finite Group DatabasebetaFinite Group Databasebeta| order | 6 |
| isoclinism | isoclinic class of D3 |
| automorphism | Aut(G) = D3 (order 6) · Inn(G) = G · Out(G) = C1 |
| holomorph | G ⋊ Aut(G) = D32 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D3 |
| general linear | GL(2,𝔽2) |
| projective special linear | PSL(2,𝔽2) |
| symmetric | S3 |
| special linear | SL(2,𝔽2) |
| projective general linear | PGL(2,𝔽2) |
| symplectic | Sp(2,𝔽2) |
| projective symplectic | PSp(2,𝔽2) |
| affine general linear | AGL(1,𝔽3) |
| general semilinear | ΓL(1,𝔽4) |
| special unitary | SU(2,𝔽2) |
| projective special unitary | PSU(2,𝔽2) |
| unambiguous semidirect product | C3⋊C2 |
| Fitting, socle, derived, Sylow | C3 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C3 | C2 |
| LCS | G ▷ C3 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 3 |
| transitive degree | μt(G) = 3 |
A minimal faithful permutation representation — 2 generators on 3 points:
(1,2,3);(2,3)Polycyclic presentation:
2,3|b^a=b^2| C1 | C2 | C3 | |
|---|---|---|---|
| size | 1 | 2 | 3 |
| order | 1 | 3 | 2 |
| χ1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | -1 |
| χ3 | 2 | -1 | 0 |
| maximal subgroup of | C2×D3, D9, C3×D3, … |
| maximal normal subgroup of | C2×D3, C3×D3, C5×D3, … |
| maximal quotient of | Dic3, C2×D3, D9, … |
| automorphism group of | C22, D3 |
| outer automorphism group of | C22, Q8, 3−1+2, … |
| holomorph of | C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(6, 1);