Alternating group A6
Linear group L2(9)
Derived symplectic group S4(2)'
Derived Mathieu group M10'

Order = 360 = 23.32.5.
Mult = 6.
Out = 22.
Robert Wilson's ATLAS page for A6 is available here.

A6: Length 25, 2-generator, 4-relator.

< x, y | x3 = y4 = (xy)5 = [x, y]2 = 1 >

Remark: This is the Coxeter group (3, 4, 5; 2).

A6: Length 31, 2-generator, 4-relator.

< x, y | x2 = y4 = (xy)5 = (xy2)5 = 1 >

Remark: x and y are R.A.Wilson's standard generators for A6.

3.A6: Length 24, 2-generator, 4-relator.

< x, y | x3 = y3 = (xy)4 = (xy-1)5 = 1 >

Remark: This is the Coxeter group (3, 3 | 4, 5).

S6: Length 42, 2-generator, 5-relator.

< x, y | x6 = y2 = (xy)5 = [x, y]3 = [x2, y]2 = 1 >

Remark: Moore/Coxeter presentation. The presentation is available in MAGMA code here. (This applies to the subgroups below too.)

Some subgroups:

Realisation: x = (1, 2, 3, 4, 5, 6) and y = (1, 2).
The above permutations are available in MAGMA format here.

3.PGL2(9): Length 41, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)8 = [x, y]5 = 1 >

Remark: This is the Coxeter group (2, 3, 8; 5).

3.PGL2(9): Length 41, 2-generator, 4-relator.

< x, y | x2 = y3 = (xy)10 = [x, y]4 = 1 >

Remark: This is the Coxeter group (2, 3, 10; 4).

M10: Length 84, 2-generator, 6-relator.

< x, y | x2 = y4 = (xy)8 = (xy2)5 = [x, y]3 = (xyxyxy2)5 = 1 >

Remark: If (g, h) are R.A.Wilson's standard generators for M10 then we have (g, h) = (x, (xy)y) and (x, y) = (g, h2ghgh4g). [A conjugate (in M10) of (x, y) is (g, hgh4).]

M10: Length 71, 2-generator, 5-relator.

< x, y | x2 = y8 = (xy3)4 = (xy4)3 = (xyxy-3)5 = 1 >

Remark: x and y are R.A.Wilson's standard generators for M10. We have o(xy) = 8 and o(xy2) = 4.

PGammaL2(9): Length 58, 2-generator, 5-relator.

< x, y | x2 = y4 = (xy)6 = [x, y]5 = (xyxy2xy2xy-1)2 = 1 >
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Last updated 23rd August, 1997
John N. Bray