Finite Group DatabasebetaFinite Group Databasebeta| order | 20 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C4×D3 (order 24) · Inn(G) = C1 · Out(G) = C4×D3 |
| holomorph | G ⋊ Aut(G) = AGL(1,𝔽5)×S4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C2 |
| abelian | C22×C5 |
| multiplicative | (ℤ/33)ˣ, (ℤ/44)ˣ, (ℤ/66)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
10 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)(5,6,7,8,9);(3,4)20 conjugacy classes — view the full character table.
| center of | C5×C22⋊C4, C5×C4⋊C4, C2×C5×D4, … |
| derived subgroup of | D5×A4, #3938, #3955, … |
| Frattini subgroup of | C4×Dic25, #87718, C4⋊Dic25, … |
| Fitting subgroup of | C2×Dic5, C22⋊D5, C22×D5, … |
| socle of | C2×Dic5, C2×C4×C5, C22×D5, … |
| maximal subgroup of | C2×Dic5, C22⋊D5, C2×C4×C5, … |
| maximal normal subgroup of | C2×Dic5, C22⋊D5, C2×C4×C5, … |
| maximal quotient of | C2×C4×C5, C5×D4, C5×Q8, … |
| automorphism group of | C3×C11, C4×C11, C2×C3×C11 |
| outer automorphism group of | C3×C11, C4×C11, C2×C3×C11, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20, 5);