Finite Group DatabasebetaFinite Group Databasebeta| order | 10 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = AGL(1,𝔽5) (order 20) · Inn(G) = G · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = D5⋊AGL(1,𝔽5) |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| dihedral | D5 |
| unambiguous semidirect product | C5⋊C2 |
| Fitting, socle, derived, Sylow | C5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 | |
4 conjugacy classes of subgroups.
| minimal degree | μ(G) = 5 |
| transitive degree | μt(G) = 5 |
A minimal faithful permutation representation — 2 generators on 5 points:
(1,2,3,4,5);(2,5)(3,4)Polycyclic presentation:
2,5|b^a=b^4| C1 | C2 | C3 | C4 | |
|---|---|---|---|---|
| size | 1 | 2 | 2 | 5 |
| order | 1 | 5 | 5 | 2 |
| χ1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | -1 |
| χ3 | 2 | 0 | ||
| χ4 | 2 | 0 |
| maximal subgroup of | AGL(1,𝔽5), C2×D5, C3×D5, … |
| maximal normal subgroup of | AGL(1,𝔽5), C2×D5, C3×D5, … |
| maximal quotient of | Dic5, C2×D5, C3×D5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(10, 1);