Finite Group DatabasebetaFinite Group Databasebeta| order | 20 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C2×C4 (order 8) · Inn(G) = C1 · Out(G) = C2×C4 |
| holomorph | G ⋊ Aut(G) = D4×AGL(1,𝔽5) |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C20 |
| abelian | C4×C5 |
| multiplicative | (ℤ/25)ˣ, (ℤ/50)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2 | C2×C5 |
| socle, prime radical | C2×C5 | C2 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 20 |
A minimal faithful permutation representation — 1 generator on 9 points:
(1,2,3,4)(5,6,7,8,9)20 conjugacy classes — view the full character table.
| center of | C5×M16, C5×#41, C5×C3⋊C8, … |
| derived subgroup of | #677, D40, Dic20, … |
| Frattini subgroup of | #6014, C8×C25, #87709, … |
| Fitting subgroup of | #168, Dic10, C4×D5, … |
| maximal subgroup of | #168, C5×C8, Dic10, … |
| maximal normal subgroup of | #168, C5×C8, Dic10, … |
| maximal quotient of | C5×C8, C2×C4×C5, C5×Dic3, … |
| automorphism group of | C25, C2×C25 |
| outer automorphism group of | C25, C2×C25, D41, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20, 2);