Finite Group DatabasebetaFinite Group Databasebeta| order | 20 |
| isoclinism | isoclinic class of D5 |
| automorphism | Aut(G) = C2×AGL(1,𝔽5) (order 40) · Inn(G) = D5 · Out(G) = C22 |
| holomorph | G ⋊ Aut(G) = #11717375 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| dicyclic | Dic5 |
| derived, Sylow | C5 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5 |
| commutator, hypocenter | C5 | C4 |
| Fitting, socle, prime radical, cosocle | C2×C5 | C2 |
| LCS | G ▷ C5 | |
| UCS | C1 ◁ C2 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 2 generators on 9 points:
(1,2)(3,4)(5,6,7,8,9);(1,3,2,4)(6,9)(7,8)Polycyclic presentation:
2^2,5|a^2=b,c^a=c^4| C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | |
|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 |
| order | 1 | 2 | 5 | 5 | 10 | 10 | 4 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 |
| χ3 | 1 | -1 | 1 | 1 | -1 | -1 | ||
| χ4 | 1 | -1 | 1 | 1 | -1 | -1 | ||
| χ5 | 2 | 2 | 0 | 0 | ||||
| χ6 | 2 | 2 | 0 | 0 | ||||
| χ7 | 2 | -2 | 0 | 0 | ||||
| χ8 | 2 | -2 | 0 | 0 |
Here i is the imaginary unit (i² = −1).
| maximal subgroup of | #170, Dic10, C4×D5, … |
| maximal normal subgroup of | #170, Dic10, C4×D5, … |
| maximal quotient of | #168, C2×Dic5, C3×Dic5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20, 1);