# Set up the group acting on 20 points:

g1 := (7,9,11,13,15)(6,8,10,12,14);
g2 := (16,19,10,4,14)(3,5,15,20,9);
g3:=(1,7,19,11,4)(20,18,12,5,6);
g4:=(5,2,9,18,13)(19,17,14,1,8);
g5:=(1,3,11,17,15)(6,2,10,18,16);
g6:=(7,2,4,13,16)(3,12,17,20,8);
grp := Group(g1,g2,g3,g4,g5,g6);
n:=Size(grp);

# Is the group transitive?
Orbits(grp,[1..20]);

#  Is the group primitive?
orb := last[1];

blocks := Blocks(grp,orb); #only use if grp is transitive

# Nope.  How about looking at the action on blocks?

op := Operation(grp,blocks,OnSets);

# Could this be A15 acting on the blocks????
Size(op);
Factorial(15);

# Seems like it might be A15!
# What is the kernel of this action and how big is it?

oph := OperationHomomorphism(grp,op);
ker := Kernel(oph);
Size(ker);
IsElementaryAbelian(ker);
# Hmmm... there seem to be 14 copies of C2 here.
# Where can we look for a complement to this kernel?
# How about the stabilizer of the set of first elements of
# the blocks?
blocks;
rep := List(blocks,x->x[1]);
rep := Set(List(blocks,x->x[1]));
stab := Stabilizer(grp,rep,OnSets);
Size(stab);
Size(grp)/(Size(stab)*Size(ker));

# It looks like stab is the right size for a complement....
Intersection(stab,ker);
Size(last);

# And there we have it, the group is a semi-direct product of A15
# acting on C2^14!