Finite Group DatabasebetaFinite Group Databasebeta| order | 120 |
| isoclinism | isoclinic class of S5 |
| automorphism | Aut(G) = S5 (order 120) · Inn(G) = G · Out(G) = C1 |
| holomorph | G ⋊ Aut(G) = S52 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| projective general linear | PGL(2,𝔽5) |
| symmetric | S5 |
| general orthogonal | GO−(4,𝔽2) |
| projective general semilinear | PΓL(2,𝔽4) |
| special semilinear | ΣL(2,𝔽4) |
| projective special semilinear | PΣL(2,𝔽4) |
| special orthogonal | SO−(4,𝔽2) |
| special orthogonal | SO(3,𝔽5) |
| unambiguous semidirect product | A5⋊C2 |
| socle, derived | A5 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, socle, cosocle, hypocenter | A5 | C2 |
| LCS | G ▷ A5 | |
| UCS | C1 | |
19 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);(1,2)| C1 | C2 | C3 | C4 | C5 | C6 | C7 | |
|---|---|---|---|---|---|---|---|
| size | 1 | 10 | 15 | 20 | 20 | 24 | 30 |
| order | 1 | 2 | 2 | 3 | 6 | 5 | 4 |
| χ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 1 | -1 | 1 | 1 | -1 | 1 | -1 |
| χ3 | 4 | 2 | 0 | 1 | -1 | -1 | 0 |
| χ4 | 4 | -2 | 0 | 1 | 1 | -1 | 0 |
| χ5 | 5 | 1 | 1 | -1 | 1 | 0 | -1 |
| χ6 | 5 | -1 | 1 | -1 | -1 | 0 | 1 |
| χ7 | 6 | 0 | -2 | 0 | 0 | 1 | 0 |
| maximal subgroup of | C2×S5, C3×S5, ΓL(2,𝔽4), … |
| maximal normal subgroup of | C2×S5, C3×S5, C5×S5, … |
| maximal quotient of | #6728, #6729, A5⋊C4, … |
| automorphism group of | A5, SL(2,𝔽5), S5, … |
| outer automorphism group of | 2−1+4, C22⋊A4, C2×C22⋊A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(120, 34);