The M₁₂-Design

A = Aut(Icosahedron) (=A₅) = < x,y > with
x = (1)(2)(3, 10, 7, 4, 11)(5, 9, 8, 12, 6),
y = (1, 4, 5, 9, 3)(2, 8, 10, 7, 6)(11)(12).

The orbits of {1, 2, 3, 4, 5, 7}, {1, 3, 4, 7, 10, 11} and {1, 4, 5, 7, 9, 10} under A form the blocks of the M₁₂-design. The stabilizer of first two sets in A is trivial, the third is of order five, generated by x. The orbits are thus of length 60, 12 and 60 respectively.

{1, 2, 3, 4, 5, 7}	{1, 3, 4, 7, 10, 11}	{1, 4, 5, 7, 9, 10}

Let i=1,...12 denote the elements of the projective line over GF(11), where we identify < (1,i) > with i , i=1,...,10 and put < (1,0) > = 11 and < (0,1) > = 12.

The matrices

6 0
0 2

and

1 9
5 2

induce the given generators x and y on the projective line. Thus, we embed A into PSL(2,11). The three orbits indicated above fuse into one orbit of length 132 under PSL(2,11). The element x generates the stabilizer of {1, 3, 4, 7, 10, 11} in PSL(2,11), too.

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Last updated: September 17, 1999, Anton Betten