Finite Group DatabasebetaFinite Group Databasebeta| order | 5 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C4 (order 4) · Inn(G) = C1 · Out(G) = C4 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C5 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
2 conjugacy classes of subgroups.
| minimal degree | μ(G) = 5 |
| transitive degree | μt(G) = 5 |
A minimal faithful permutation representation — 1 generator on 5 points:
(1,2,3,4,5)| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 |
| order | 1 | 5 | 5 | 5 | 5 |
| χ1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | ζ | ζ2 | ζ3 | ζ4 |
| χ3 | 1 | ζ2 | ζ4 | ζ | ζ3 |
| χ4 | 1 | ζ3 | ζ | ζ4 | ζ2 |
| χ5 | 1 | ζ4 | ζ3 | ζ2 | ζ |
Here ζ is a primitive root of unity of order 5 (ζ5 = 1).
| center of | C5×D3, C5×D5, C5×A4, … |
| derived subgroup of | D5, Dic5, AGL(1,𝔽5), … |
| Frattini subgroup of | C25, D25, C2×C25, … |
| Fitting subgroup of | D5, AGL(1,𝔽5), C5×A5, … |
| socle of | D5, AGL(1,𝔽5), C25, … |
| maximal subgroup of | D5, C2×C5, C3×C5, … |
| maximal normal subgroup of | D5, C2×C5, C3×C5, … |
| maximal quotient of | C2×C5, C3×C5, C25, … |
| outer automorphism group of | D11, #1035, D3×D11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(5, 1);