Two new symmetric 2-(144,66,30) designs that can be constructed as
follows:

Take a subgroup L = L_2(11) in the Mathieu group M_12 which acts
intransitively on 12 points, i.e. L is not maximal in M_12.

L induces five orbits O_1, ..., O_5 on the set O of its own cosets in
M_12 , where

|O_1| = 1, |O_2| = |O_3| = 11 , |O_4| = 55 and  |O_5| = 66.

Now take  B_1 = O_5^M_12 ,
          B_2 = (O_2 join O_4)^M_12  and
          B_3 = (O_3 join O_4)^M_12  as well as
          D_i = ( O , B_i)   for  i = 1,2,3 .

Then D_i is a symmetric 2-(144,66,30) design for i = 1,2,3 .

Moreover,  Aut(D_1) = Aut(M_12)  and
           Aut(D_2) = Aut(D_3) = M_12 ,
where an outer automorphism of  M_12  induces an isomorphism
between  D_2 and D_3 .

The designs are not isomorphic to the one found previously
by W. Wirth  by taking  L  to be a maximal  L_2(11)  of M_12 ,
because the 3-ranks of the corresponding incidence matrices
are mutually different.



Wolfgang Lempken, Dec. 1999