Alternating group A₅
Linear group L₂(4)
Linear group L₂(5)

Order = 60 = 2².3.5.
Mult = 2.
Out = 2.

Robert Wilson's ATLAS page for A₅ is available here.

A₅: Length 15, 2-generator, 3-relator.

< x, y | x² = y³ = (xy)⁵ = 1 >

Remark: This is the Coxeter group (2, 3, 5).

A₅: Length 12, 2-generator, 3-relator.

< x, y | x⁵ = y³ = (xy)² = 1 >

Remark: This is the Coxeter group (2, 3, 5).

A₅: Length 41, 3-generator, 7-relator.

< a, b, c | a² = b² = c² = (ab)³ = (ac)² = (bc)⁵ = (abc)⁵ = 1 >

Remark: This is the Coxeter group G^3,5,5.

2.A₅: Length 13, 2-generator, 2-relator.

< x, y | x⁵y^-3 = xyxy^-2 = 1 >
otherwise written as:
< x, y | x⁵ = y³ = (xy)² >

Remark: This is the Coxeter group <2, 3, 5>.

S₅: Length 28, 2-generator, 4-relator.

< x, y | x² = y⁴ = (xy)⁵ = [x, y]³ = 1 >

Remark: This is the Coxeter group (2, 4, 5; 3). x and y are R.A.Wilson's standard generators for S₅.
Reciprocity: (x, y) and (x, y^-1) are conjugate in G = Aut(G).

S₅: Length 27, 2-generator, 4-relator.

< x, y | x² = y⁵ = (xy)⁶ = [x, y]² = 1 >

Remark: This is the Coxeter group (2, 5, 6; 2). We may replace (xy)⁶ = 1 by (xy²)⁴ = 1. R.A.Wilson's standard generators for S₅ are x and xy².
Reciprocity: (x, y) and (x, y^-1) are conjugate in G = Aut(G), but they are not conjugate (in G) to (x, y²) and (x, y^-2).

A₅ × 2: Length 26, 3-generator, 6-relator.

< a, b, c | a² = b² = c² = (ab)³ = (ac)² = (bc)⁵ = 1 >

Remark: This is the Coxeter group G^3,5,10. The `10' is redundant and may be replaced by `infinity' or `0'.

Last updated 23^rd August, 1997

John N. Bray

Alternating group A5 Linear group L2(4) Linear group L2(5)

A5: Length 15, 2-generator, 3-relator.

A5: Length 12, 2-generator, 3-relator.

A5: Length 41, 3-generator, 7-relator.

2.A5: Length 13, 2-generator, 2-relator.

S5: Length 28, 2-generator, 4-relator.

S5: Length 27, 2-generator, 4-relator.

A5 × 2: Length 26, 3-generator, 6-relator.