Finite Group DatabasebetaFinite Group Databasebeta| order | 60 |
| isoclinism | isoclinic class of A5 |
| automorphism | Aut(G) = S5 (order 120) · Inn(G) = G · Out(G) = C2 |
| holomorph | G ⋊ Aut(G) = #72219548 |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C2 |
| alternating | A5 |
| projective special linear | PSL(2,𝔽4) |
| projective special linear | PSL(2,𝔽5) |
| special linear | SL(2,𝔽4) |
| projective general linear | PGL(2,𝔽4) |
| symplectic | Sp(2,𝔽4) |
| projective symplectic | PSp(2,𝔽5) |
| projective symplectic | PSp(2,𝔽4) |
| omega | Ω−(4,𝔽2) |
| omega | Ω(3,𝔽5) |
| special unitary | SU(2,𝔽4) |
| projective special unitary | PSU(2,𝔽4) |
| projective special unitary | PSU(2,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, hypocenter | G | C1 |
| LCS | G | |
| UCS | C1 | |
9 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);(1,2,3)| C1 | C2 | C3 | C4 | C5 | |
|---|---|---|---|---|---|
| size | 1 | 12 | 12 | 15 | 20 |
| order | 1 | 5 | 5 | 2 | 3 |
| χ1 | 1 | 1 | 1 | 1 | 1 |
| χ2 | 3 | -1 | 0 | ||
| χ3 | 3 | -1 | 0 | ||
| χ4 | 4 | -1 | -1 | 0 | 1 |
| χ5 | 5 | 0 | 0 | 1 | -1 |
| derived subgroup of | S5, C2×A5, C3×A5, … |
| socle of | S5 |
| maximal subgroup of | S5, C2×A5, C3×A5, … |
| maximal normal subgroup of | S5, C2×A5, C3×A5, … |
| maximal quotient of | SL(2,𝔽5), C2×A5, C3×A5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(60, 5);