Dear Robert,
here is essentially how I did it:
(The generators this produces are different from the ones
I sent earlier, because the details of the constructions differ.)
#####
# finds generators of 2x2^3:L3(2) inside M22.2
slp:=AtlasStraightLineProgram("M22.2",5).program;
# 3.M22.2 in dimension 12, std.gens. lifting those of M22.2
gens:=AtlasGenerators("3.M22.2",1).generators;
# a subgroup of 3.(2x2^3:L3(2)) projecting onto 2x2^3:L3(2)
hgens:=ResultOfStraightLineProgram(slp,gens);
# indeed 2x2^3:L3(2)
g:=Group(gens); h:=Group(hgens); Size(h);
# go over to perm.rep, actually on 693 points
iso:=IsomorphismPermGroup(g); LargestMovedPointPerm(gg);
gg:=Image(iso,g); hh:=Image(iso,h);
# action on cosets
cos:=RightCosets(gg,hh); Length(cos);
act:=Action(gg,cos,OnRight); LargestMovedPointPerm(act);
#####
Best wishes, Jürgen
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