|
#setup: extraspecial group r^(1+2n) in GL(r^n,q), r prime, r|q-1
#
#the functions in this file:
#
# extraspecial(r,n,q)
# output: the extraspecial group r^(1+2n)
# (in the case r=2, the output is + type)
#
# minusextraspecial(r,n,q)
# output: the group 2^(1+2n), of - type
#
# c6(r,n,q)
# output: the normalizer of r^(1+2n) in GL(r^n,q)
# (in the case r=2, the extraspecial is + type
#
# minusc6(r,n,q)
# output: the normalizer of 2^(1+2n) of - type in GL(r^n,q)
#
#the groups in this file: maximal subgroups of Sp(4,p) for p=5,11,13
#as 4-dimensional groups, in lists maximals5, maximals11, maximals13
#constructs r^(1+2n) in GL(r^n,q)
#when r=2, the group is + type (central product of dihedrals)
extraspecial:=function(r,n,q)
local a,b,rho,i,perm,id,id2,gens;
a:=NullMat(r,r,GF(q));
rho:=Z(q)^((q-1)/r);
for i in [1..r] do
a[i][i]:=rho^(i-1);
od;
perm:=Concatenation([2..r],[1]);
perm:=PermList(perm);
b:=PermutationMat(perm,r,GF(q));
id:=a^0;
gens:=[a,b];
for i in [1..n-1] do
id2:=gens[1]^0;
gens:=List(gens,x->KroneckerProduct(x,id));
Add(gens,KroneckerProduct(id2,a));
Add(gens,KroneckerProduct(id2,b));
od;
return Group(gens);
end;
#constructs the group 2^(1+2n) of minus type
minusextraspecial:=function(r,n,q)
local a,b,c,d,mark,x,y,diff,elmts,i,j,id,id2,gens;
c:=[ [0*Z(q),-Z(q)^0], [Z(q)^0, 0*Z(q)] ];
#this is really inefficient
diff:=Difference(Elements(GF(q)),[0*Z(q)]);
elmts:=Elements(GF(q));
mark:=false;
j:=0;
repeat
j:=j+1;
x:=diff[j];
i:=0;
repeat
i:=i+1;
y:=elmts[i];
if x^2+y^2=-Z(q)^0
then mark:=true;
fi;
until mark or i=q;
until j=q-1 or mark;
d:=[ [x,y], [y,-x] ];
if n=1 then
return Group(c,d);
else
a:=[ [Z(q)^0, 0*Z(q)], [0*Z(q),-Z(q)^0] ];
b:=[ [0*Z(q), Z(q)^0], [Z(q)^0, 0*Z(q)] ];
id:=a^0;
gens:=[a,b];
for i in [1..n-2] do
id2:=gens[1]^0;
gens:=List(gens,x->KroneckerProduct(x,id));
Add(gens,KroneckerProduct(id2,a));
Add(gens,KroneckerProduct(id2,b));
od;
id2:=gens[1]^0;
gens:=List(gens,x->KroneckerProduct(x,id));
Add(gens,KroneckerProduct(id2,c));
Add(gens,KroneckerProduct(id2,d));
return Group(gens);
fi;
end;
#constructs the normalizer of r^(1+2n) in GL(r^n,q)
#when r=2, the top group is O^+(2n,2)
c6:=function(r,n,q)
local g,gens,i,j,rho,a,b,c,id,id2,id3,gens2;
g:=extraspecial(r,n,q);
gens:=[];
for i in [1..2*n] do
gens[i]:=GeneratorsOfGroup(g)[i];
od;
a:=NullMat(r,r,GF(q));
rho:=Z(q)^((q-1)/r);
for i in [1..r] do
a[i][i]:=rho^(i*(i-1)/2);
od;
b:=List([1..r],x->[]);
for i in [1..r] do
for j in [1..r] do
b[i][j]:=rho^((i-1)*(j-1));
od;
od;
c:=NullMat(r^2,r^2,GF(q));
for i in [0..r^2-1] do
c[i+1][((i+( (i-1) mod r )*r) mod r^2)+1]:=Z(q)^0;
od;
id:=IdentityMat(r,GF(q));
gens2:=[a,b];
for i in [1..n-1] do
gens2:=List(gens2,x->KroneckerProduct(x,id));
id2:=IdentityMat(r^i,GF(q));
Add(gens2,KroneckerProduct(id2,a));
Add(gens2,KroneckerProduct(id2,b));
id3:=IdentityMat(r^(i-1),GF(q));
Add(gens2,KroneckerProduct(id3,c));
od;
return Group(Concatenation(gens,gens2));
end;
#constructs the normalizer of 2^(1+2n) of - type in GL(2^n,q)
minusc6:=function(r,n,q)
local diff,elmts,mark,x,y,g,gens,c,d,e,u,v,w,i,j,id,id2,id3;
c:=[ [0*Z(q),-Z(q)^0], [Z(q)^0, 0*Z(q)] ];
#this is really inefficient
diff:=Difference(Elements(GF(q)),[0*Z(q)]);
elmts:=Elements(GF(q));
mark:=false;
j:=0;
repeat
j:=j+1;
x:=diff[j];
i:=0;
repeat
i:=i+1;
y:=elmts[i];
if x^2+y^2=-Z(q)^0
then mark:=true;
fi;
until mark or i=q;
until j=q-1 or mark;
d:=[ [x,y], [y,-x] ];
e:=Z(q)^0;
u:=[ [e,e], [-e,e] ];
v:=[ [ e+x+y, e-x+y ], [ -e-x+y, e-x-y ] ];
if n=1 then return Group(c,d,u,v); fi;
w:=[ [e,0*e,e,0*e], [0*e,e,0*e,e], [0*e,e,0*e,-e], [-e,0*e,e,0*e] ];
g:=c6(r,n-1,q);
id:=IdentityMat(r,GF(q));
id2:=IdentityMat(r^(n-1),GF(q));
id3:=IdentityMat(r^(n-2),GF(q));
gens:=List(GeneratorsOfGroup(g),z->KroneckerProduct(z,id));
Add(gens,KroneckerProduct(id2,c));
Add(gens,KroneckerProduct(id2,d));
Add(gens,KroneckerProduct(id2,u));
Add(gens,KroneckerProduct(id2,v));
Add(gens,KroneckerProduct(id3,w));
return(Group(gens));
end;
f := GF(5);
maximals5 := [
Group([ [
[3,4,3,1],
[0,2,3,3],
[2,0,1,1],
[3,2,0,0] ]*One(f), [
[0,3,3,2],
[1,3,2,3],
[1,0,4,2],
[4,1,4,2] ]*One(f), [
[0,2,2,4],
[3,0,1,3],
[3,4,0,3],
[1,3,3,2] ]*One(f), [
[0,0,4,4],
[1,4,1,3],
[0,0,4,0],
[1,0,3,1] ]*One(f), [
[1,1,3,2],
[3,3,2,3],
[4,2,0,4],
[3,4,2,2] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[0,0,1,1],
[4,1,3,0],
[0,0,1,0],
[4,0,0,0] ]*One(f) ] ),
Group([ [
[3,4,3,1],
[0,2,3,3],
[2,0,1,1],
[3,2,0,0] ]*One(f), [
[0,2,2,4],
[3,0,1,3],
[3,4,0,3],
[1,3,3,2] ]*One(f), [
[1,1,3,2],
[3,3,2,3],
[4,2,0,4],
[3,4,2,2] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[3,0,4,2],
[3,4,3,4],
[0,0,4,0],
[2,0,2,0] ]*One(f), [
[1,4,4,1],
[2,1,3,4],
[2,1,0,1],
[3,2,3,0] ]*One(f), [
[0,0,1,1],
[4,1,3,0],
[0,0,1,0],
[4,0,0,0] ]*One(f) ]),
Group([ [
[0,1,1,1],
[3,1,4,2],
[4,4,3,4],
[3,1,0,0] ]*One(f), [
[0,1,2,0],
[3,0,0,3],
[4,0,0,1],
[0,1,3,0] ]*One(f), [
[1,1,3,1],
[3,1,4,2],
[3,4,1,4],
[0,1,4,0] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f) ] ),
Group([ [
[0,2,2,4],
[3,0,1,3],
[3,4,0,3],
[1,3,3,2] ]*One(f), [
[4,2,2,3],
[4,1,3,2],
[0,0,4,3],
[0,0,1,1] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[1,4,4,1],
[2,1,3,4],
[2,1,0,1],
[3,2,3,0] ]*One(f) ] ),
Group([ [
[1,1,1,4],
[3,0,1,1],
[3,3,1,4],
[2,3,2,0] ]*One(f), [
[3,4,1,0],
[0,2,0,4],
[2,0,2,4],
[0,3,0,3] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[0,3,2,1],
[3,3,3,2],
[2,3,1,2],
[1,2,2,4] ]*One(f) ] ),
Group([ [
[1,1,1,4],
[1,1,4,1],
[1,4,4,4],
[4,1,4,4] ]*One(f), [
[4,3,2,1],
[3,3,2,2],
[2,0,3,2],
[4,2,2,2] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[0,1,2,3],
[3,1,1,3],
[4,2,1,3],
[2,3,3,2] ]*One(f) ]),
Group([ [
[2,1,4,4],
[0,4,3,4],
[3,4,0,4],
[4,3,0,2] ]*One(f), [
[3,2,3,4],
[4,0,3,0],
[4,4,1,2],
[4,4,0,3] ]*One(f), [
[2,3,3,2],
[2,1,0,1],
[3,3,3,3],
[3,3,4,2] ]*One(f), [
[2,3,4,3],
[2,3,0,4],
[3,1,3,2],
[3,3,3,4] ]*One(f), [
[0,4,3,0],
[4,0,0,2],
[0,0,0,4],
[0,0,4,0] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f), [
[4,2,4,2],
[3,3,1,4],
[0,4,2,3],
[1,0,2,1] ]*One(f), [
[4,1,4,2],
[0,3,1,4],
[3,3,2,4],
[3,3,0,1] ]*One(f), [
[0,0,4,3],
[4,2,2,4],
[3,4,3,0],
[4,3,1,0] ]*One(f) ] ),
Group([ [
[2,3,1,1],
[1,4,1,1],
[4,4,0,4],
[4,2,4,0] ]*One(f), [
[3,2,3,3],
[0,0,2,2],
[3,4,3,0],
[4,1,1,3] ]*One(f), [
[4,0,0,0],
[0,4,0,0],
[0,0,4,0],
[0,0,0,4] ]*One(f) ] )
];
magmasp45:=Group(
[ [2, 0, 0, 0],
[0, 1, 0, 0],
[0, 0, 1, 0],
[0, 0, 0, 3]] *One(f),
[ [1, 0, 1, 0],
[1, 0, 0, 0],
[0, 1, 0, 1],
[0, 4, 0, 0] ]*One(f));
f := GF(7);
maximals7 := [
Group([[
[4,3,1,0],
[2,6,0,5],
[1,0,6,6],
[2,3,1,2]
]*One(f),
[
[1,0,1,5],
[3,3,4,4],
[2,2,3,5],
[2,0,1,4]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[4,4,2,2],
[5,6,0,2],
[4,0,6,3],
[5,4,2,1]
]*One(f),
[
[2,3,5,6],
[2,0,3,5],
[3,2,2,4],
[1,3,5,0]
]*One(f),
[
[2,2,3,0],
[6,3,5,3],
[5,5,6,5],
[2,5,1,0]
]*One(f),
[
[5,3,1,1],
[0,0,2,3],
[0,3,2,5],
[4,1,5,1]
]*One(f)]),
Group([[
[4,3,1,0],
[2,6,0,5],
[1,0,6,6],
[2,3,1,2]
]*One(f),
[
[1,0,1,5],
[3,3,4,4],
[2,2,3,5],
[2,0,1,4]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[6,0,0,0],
[3,5,2,0],
[5,3,0,0],
[6,5,4,6]
]*One(f),
[
[2,3,5,6],
[2,0,3,5],
[3,2,2,4],
[1,3,5,0]
]*One(f),
[
[2,2,3,0],
[6,3,5,3],
[5,5,6,5],
[2,5,1,0]
]*One(f),
[
[2,4,0,4],
[1,0,1,0],
[6,2,1,3],
[0,6,6,6]
]*One(f)]) ,
Group([[
[0,0,2,0],
[2,0,0,5],
[4,0,0,0],
[0,3,2,0]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[0,4,6,1],
[3,6,1,2],
[6,0,5,6],
[5,3,6,4]
]*One(f),
[
[4,1,5,2],
[5,4,0,4],
[4,2,5,5],
[0,5,6,5]
]*One(f)]),
Group([[
[4,0,3,2],
[2,5,2,3],
[4,2,2,0],
[3,4,5,3]
]*One(f),
[
[1,0,1,5],
[3,3,4,4],
[2,2,3,5],
[2,0,1,4]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[2,4,0,4],
[1,0,1,0],
[6,2,1,3],
[0,6,6,6]
]*One(f)]),
Group([[
[2,2,3,5],
[3,2,4,3],
[0,4,1,5],
[6,0,4,1]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[0,2,2,3],
[4,3,2,2],
[6,1,3,5],
[0,6,3,6]
]*One(f),
[
[6,2,4,0],
[4,1,0,3],
[5,0,1,2],
[0,2,4,6]
]*One(f)]),
Group([[
[4,1,0,5],
[2,6,3,0],
[5,1,1,6],
[6,5,5,3]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[1,2,4,6],
[6,1,6,0],
[6,2,5,6],
[6,6,2,1]
]*One(f),
[
[3,2,4,3],
[4,4,1,4],
[2,6,4,5],
[4,2,3,5]
]*One(f)]),
Group([[
[5,3,3,0],
[5,2,0,4],
[1,0,2,3],
[0,6,5,5]
]*One(f),
[
[2,2,6,4],
[1,6,3,3],
[2,2,2,1],
[3,1,4,2]
]*One(f),
[
[5,2,4,1],
[2,4,6,4],
[5,1,6,5],
[0,5,5,5]
]*One(f),
[
[0,1,2,4],
[1,3,3,2],
[6,4,4,6],
[0,6,6,0]
]*One(f),
[
[3,2,5,5],
[0,3,0,5],
[5,1,4,5],
[0,5,0,4]
]*One(f),
[
[4,0,2,2],
[5,5,2,5],
[5,3,1,4],
[2,6,4,6]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[5,5,2,0],
[2,2,0,5],
[4,0,2,5],
[0,3,2,5]
]*One(f),
[
[3,5,2,6],
[6,5,4,4],
[4,4,2,5],
[4,3,4,6]
]*One(f),
[
[5,5,3,6],
[5,6,1,3],
[2,6,4,2],
[0,2,2,5]
]*One(f),
[
[0,6,2,2],
[5,2,0,2],
[1,3,6,1],
[1,1,2,1]
]*One(f),
[
[4,3,6,4],
[3,4,4,6],
[4,5,3,4],
[5,4,4,3]
]*One(f),
[
[1,3,5,0],
[3,6,0,2],
[1,0,6,3],
[0,6,3,1]
]*One(f)]),
Group([[
[6,1,6,4],
[0,4,2,6],
[2,5,4,1],
[6,0,2,5]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[4,2,0,1],
[2,0,1,0],
[0,2,4,5],
[2,0,5,0]
]*One(f),
[
[1,6,4,6],
[3,0,0,4],
[4,3,0,1],
[1,4,4,6]
]*One(f),
[
[5,6,3,2],
[2,1,1,3],
[6,1,0,3],
[5,4,0,6]
]*One(f),
[
[6,2,1,0],
[2,1,0,6],
[3,0,1,2],
[0,4,2,6]
]*One(f),
[
[2,5,0,0],
[5,5,0,0],
[5,0,5,5],
[0,2,5,2]
]*One(f),
[
[2,0,0,3],
[4,6,6,6],
[6,4,3,2],
[3,3,5,5]
]*One(f),
[
[6,6,5,0],
[6,1,0,2],
[4,0,1,6],
[0,3,6,6]
]*One(f),
[
[0,4,1,5],
[5,3,0,1],
[1,1,5,3],
[4,1,2,1]
]*One(f),
[
[1,1,0,4],
[1,6,4,0],
[0,1,1,6],
[1,0,6,6]
]*One(f),
[
[1,1,6,6],
[0,4,4,6],
[6,6,3,6],
[3,6,0,6]
]*One(f),
[
[0,5,0,6],
[5,4,6,0],
[0,5,0,2],
[5,0,2,4]
]*One(f)]),
Group([[
[2,1,4,6],
[5,0,0,1],
[5,2,2,4],
[3,0,1,2]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[3,1,5,0],
[4,6,0,3],
[2,0,6,5],
[5,1,5,5]
]*One(f)]),
Group([[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[4,0,6,3],
[1,4,0,5],
[2,3,3,6],
[6,3,3,4]
]*One(f),
[
[1,3,4,5],
[6,2,3,1],
[3,1,1,2],
[1,3,0,4]
]*One(f)]),
Group([[
[2,0,0,2],
[6,1,2,6],
[1,4,2,1],
[4,6,2,1]
]*One(f),
[
[6,0,0,0],
[0,6,0,0],
[0,0,6,0],
[0,0,0,6]
]*One(f),
[
[5,0,0,0],
[0,5,0,0],
[0,5,3,0],
[2,0,0,3]
]*One(f)])];
f := GF(11);
maximals11 := [
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[7,8,3,7],
[4,5,3,9],
[1,5,7,3],
[7,9,1,2]
]*One(f),
[
[6,9,9,1],
[1,5,4,9],
[10,7,8,2],
[8,10,10,7]
]*One(f),
[
[9,9,2,8],
[6,8,4,8],
[8,5,5,2],
[0,5,10,1]
]*One(f),
[
[7,3,7,1],
[7,8,1,7],
[2,5,1,8],
[0,2,4,2]
]*One(f),
[
[9,8,1,1],
[4,0,7,4],
[2,2,4,3],
[5,6,4,10]
]*One(f),
[
[10,9,1,3],
[6,2,6,1],
[9,4,0,2],
[7,9,5,3]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[7,8,3,7],
[4,5,3,9],
[1,5,7,3],
[7,9,1,2]
]*One(f),
[
[6,9,9,1],
[1,5,4,9],
[10,7,8,2],
[8,10,10,7]
]*One(f),
[
[9,9,2,8],
[6,8,4,8],
[8,5,5,2],
[0,5,10,1]
]*One(f),
[
[7,3,7,1],
[7,8,1,7],
[2,5,1,8],
[0,2,4,2]
]*One(f),
[
[1,2,4,3],
[0,0,9,4],
[0,6,2,9],
[0,0,0,1]
]*One(f),
[
[2,5,3,3],
[7,2,2,3],
[5,1,10,6],
[8,5,4,10]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[6,0,3,8],
[3,0,4,4],
[7,4,5,4],
[0,9,4,1]
]*One(f),
[
[9,6,8,7],
[0,8,4,2],
[7,5,3,3],
[7,6,2,1]
]*One(f),
[
[10,4,6,0],
[8,1,0,5],
[2,0,1,4],
[0,9,8,10]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[7,8,3,7],
[4,5,3,9],
[1,5,7,3],
[7,9,1,2]
]*One(f),
[
[2,5,3,3],
[7,2,2,3],
[5,1,10,6],
[8,5,4,10]
]*One(f),
[
[9,5,10,9],
[6,7,9,10],
[1,1,4,6],
[6,1,5,2]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[5,1,1,0],
[3,3,9,1],
[1,9,3,10],
[7,1,8,1]
]*One(f),
[
[1,10,10,0],
[1,10,0,1],
[10,0,10,10],
[0,1,1,1]
]*One(f),
[
[8,2,4,2],
[10,10,3,4],
[3,0,0,9],
[8,3,1,2]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[9,5,10,8],
[2,3,3,1],
[8,2,0,2],
[8,3,3,4]
]*One(f),
[
[10,5,6,5],
[3,2,0,6],
[8,8,10,6],
[0,8,8,2]
]*One(f),
[
[9,6,9,7],
[8,2,8,9],
[6,10,9,5],
[2,6,3,2]
]*One(f)]),
Group([[
[6,1,8,8],
[7,9,5,4],
[5,1,6,7],
[3,5,3,3]
]*One(f),
[
[7,9,3,3],
[2,10,9,3],
[2,2,1,2],
[1,2,9,4]
]*One(f),
[
[9,9,8,7],
[1,8,3,0],
[4,5,4,6],
[3,6,6,3]
]*One(f),
[
[0,7,3,0],
[4,0,0,8],
[2,0,0,7],
[0,9,4,0]
]*One(f),
[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[7,5,7,0],
[2,1,7,7],
[3,3,9,6],
[7,3,9,3]
]*One(f),
[
[8,8,4,0],
[10,3,0,7],
[0,0,3,8],
[0,0,10,8]
]*One(f),
[
[2,4,9,7],
[9,7,9,1],
[5,2,5,0],
[2,7,9,10]
]*One(f),
[
[10,9,6,9],
[8,1,2,6],
[1,4,10,2],
[7,1,3,1]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[4,9,9,1],
[1,3,6,10],
[6,7,8,2],
[6,8,4,8]
]*One(f),
[
[3,0,1,4],
[5,4,9,8],
[5,2,1,0],
[1,7,10,4]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[8,10,7,0],
[0,3,0,4],
[2,0,3,10],
[0,9,0,8]
]*One(f),
[
[7,3,10,3],
[10,6,5,6],
[0,1,8,9],
[5,7,2,2]
]*One(f)]),
Group([[
[10,0,0,0],
[0,10,0,0],
[0,0,10,0],
[0,0,0,10]
]*One(f),
[
[4,9,4,0],
[9,7,0,7],
[9,0,7,9],
[0,2,9,4]
]*One(f),
[
[9,7,7,4],
[6,2,7,0],
[10,1,6,3],
[5,3,4,4]
] * One(f)])];
f := GF(13);
maximals13 := [
Group([[
[7,0,7,3],
[3,2,3,7],
[2,1,9,0],
[0,2,10,4]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[12,10,6,1],
[9,7,7,6],
[6,1,8,3],
[0,6,4,3]
]*One(f),
[
[1,1,5,9],
[0,3,10,5],
[0,10,12,12],
[0,0,0,1]
]*One(f),
[
[4,9,7,8],
[3,1,1,12],
[12,5,4,1],
[6,3,4,2]
]*One(f),
[
[11,11,0,9],
[7,6,6,8],
[9,11,0,5],
[0,5,12,7]
]*One(f),
[
[10,10,6,2],
[0,9,11,10],
[9,1,7,11],
[8,7,3,12]
]*One(f)]),
Group([[
[3,7,4,3],
[2,3,12,4],
[2,9,9,6],
[10,2,11,9]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[12,10,6,1],
[9,7,7,6],
[6,1,8,3],
[0,6,4,3]
]*One(f),
[
[1,1,5,9],
[0,3,10,5],
[0,10,12,12],
[0,0,0,1]
]*One(f),
[
[4,9,7,8],
[3,1,1,12],
[12,5,4,1],
[6,3,4,2]
]*One(f),
[
[2,10,12,9],
[4,8,3,12],
[1,12,3,3],
[12,1,9,9]
]*One(f),
[
[11,11,0,9],
[7,6,6,8],
[9,11,0,5],
[0,5,12,7]
]*One(f)]),
Group([[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[7,11,9,9],
[1,9,9,8],
[12,7,1,8],
[5,0,5,8]
]*One(f),
[
[10,9,0,9],
[7,9,2,11],
[12,9,0,3],
[12,1,5,9]
]*One(f),
[
[5,3,0,0],
[5,8,0,0],
[5,0,8,3],
[0,8,5,5]
]*One(f)]),
Group([[
[3,7,4,3],
[2,3,12,4],
[2,9,9,6],
[10,2,11,9]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[2,2,10,1],
[6,7,2,10],
[2,11,6,11],
[2,2,7,11]
]*One(f),
[
[11,11,0,9],
[7,6,6,8],
[9,11,0,5],
[0,5,12,7]
]*One(f)]),
Group([[
[3,3,12,8],
[5,11,2,12],
[3,3,10,10],
[10,3,8,5]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[12,6,4,0],
[9,1,0,9],
[6,0,1,6],
[0,7,9,12]
]*One(f),
[
[12,10,11,7],
[5,8,3,11],
[12,7,6,3],
[7,12,8,2]
]*One(f)]),
Group([[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[12,7,5,6],
[6,3,12,7],
[4,0,5,11],
[6,11,4,1]
]*One(f),
[
[0,2,5,5],
[9,6,5,5],
[11,0,8,11],
[6,11,4,1]
]*One(f),
[
[12,2,7,0],
[6,6,5,7],
[11,6,7,11],
[12,11,7,1]
]*One(f)]),
Group([[
[12,7,5,2],
[5,7,7,4],
[3,1,12,8],
[4,8,6,6]
]*One(f),
[
[11,6,4,4],
[3,1,4,4],
[1,7,12,7],
[3,1,10,2]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[10,8,1,1],
[4,0,9,1],
[10,1,0,5],
[0,10,9,3]
]*One(f),
[
[4,0,0,0],
[5,1,10,0],
[11,9,0,0],
[8,11,8,10]
]*One(f),
[
[3,5,12,12],
[10,10,7,12],
[1,12,3,8],
[7,1,3,10]
]*One(f),
[
[1,0,2,7],
[11,1,5,1],
[7,9,5,2],
[10,12,0,4]
]*One(f),
[
[7,11,12,0],
[2,6,0,1],
[5,0,6,11],
[0,8,2,7]
]*One(f),
[
[9,2,10,10],
[7,2,0,10],
[10,1,12,11],
[6,10,6,5]
]*One(f)]),
Group([[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[2,9,5,0],
[12,12,0,4],
[5,1,3,2],
[6,12,10,5]
]*One(f),
[
[4,9,10,6],
[1,3,9,5],
[6,10,12,7],
[0,2,4,6]
]*One(f)]),
Group([[
[1,9,12,0],
[0,12,0,1],
[0,0,12,9],
[0,0,0,1]
]*One(f),
[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[10,12,4,8],
[2,5,10,2],
[6,5,0,12],
[9,5,7,10]
]*One(f)]),
Group([[
[12,0,0,0],
[0,12,0,0],
[0,0,12,0],
[0,0,0,12]
]*One(f),
[
[11,8,10,0],
[1,2,0,3],
[8,0,2,8],
[0,5,1,11]
]*One(f),
[
[7,2,11,8],
[10,0,12,8],
[12,1,3,7],
[0,6,7,2]
]*One(f)] )];
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