#We find all primitive subgroups of $GL(36,2)$ with parameters $e=9$, $a=4$. Since $b=2$ divides $a$, we use item (i) of Lemma 5.5. We start with primitive #subgroups in $GL(9,4)$, and then produce primitive subgroups in $GL(36,2)$. #In the notations of Theorem 5.1, if $H$ is a primitive solvable subgroup, then it possesses a series #$$1 e:=9;; p:=2;; d:=9;; a:=4;; #We apply Lemma 5.5 to construct primitive solvable subgroups of GL(36,2), so we start with primitive solvable subgroups in $GL(e,p^b)=GL(9,2^2)$ gap> l0:=AllIrreducibleSolvableMatrixGroups(Degree,d,Field,GF(p^2),IsPrimitiveMatrixGroup,true);; #In this list we choose subgroups, whose order is divisible by the order of $|F|=(2^2-1)*9*9$ gap> l1:=Filtered(l0,x-> (Order(x) mod ((p^2-1)*e*e)) = 0);; #In view of Lemma 5.4, we choose subgroups satisfying $\vert A/F\vert\in \{40, 320, 192,1152\}$ and collect these subgroups in the list l2. #Since in $GL(9,2^2)$ we have $A=H$, we have $|H|\in \{40, 320, 192,1152\}$ gap> l2:=Filtered(l1,x-> (Order(x)/((p^2-1)*e*e)) in [40, 320, 192, 1152]);; gap> Size(l2); 7 #Now we need to make an embedding of the primitive solvable subgroups of GL(9,2^2) into GL(36,2). We do it in two steps. First each $G\leq GL(9,2^2)$ we embed in #$GL(18,4)$ by taking $I_2\otimes G$, then we substitute each entry $g_{i,j}\in GF(2^2)$ of every $g\in G$ by a 2\times2-matrix over $GF(2)$ #First we produce generators of subgroups from the list l2 gap> gens:=[];; gap> for j in l2 do > Append(gens,[GeneratorsOfGroup(j)]);; > od; #Each matrix $g$ in gens we substitute by $I_2\otimes g$. gap> genss:=[];; gap> for i in [1..Size(gens)] do > genss[i]:=[]; > for j in gens[i] do > Append(genss[i],[KroneckerProduct(IdentityMat(2,Z(2)),j)]); > od; > od; #Now we produce generators of these subgroups in GL(36,2) by substituting each entry $g_{i,j}\in GF(2^2)$ of every $g\in G$ by a 2\times2-matrix over $GF(2)$ gap> bas2:= Basis(AsVectorSpace( GF(p),GF(p^2) ) );; gap> genS:=[];; gap> for i in [1..Size(genss)] do > genS[i]:=[];; > for j in genss[i] do > Append(genS[i],[BlownUpMat(bas2,j)]); > od; > od; #Now we produce $t,s\in GL(2,2^2)$, then $t\otimes I_9$, $s\otimes I_9$ in $GL(18,2^2)$, and then we produce generators of these subgroups in GL(36,2) by #substituting each entry $g_{i,j}\in GF(2^2)$ of every $g\in G$ by a 2\times2-matrix over $GF(2)$ gap> bas4:= Basis(AsVectorSpace( GF(p^2),GF(p^a) ) );; gap> t:=BlownUpMat(bas4,[[Z(p^a)]]);; gap> s:=RepresentativeAction(GL(2,p^2),t,t^(p^2));; gap> Order(s); 2 gap> I9:=IdentityMat(9,GF(2));; gap> t18:=KroneckerProduct(t,I9);; gap> s18:=KroneckerProduct(s,I9);; gap> t36:=BlownUpMat(bas2,t18);; gap> s36:=BlownUpMat(bas2,s18);; #We element t36 and s36 to generators genS gap> for i in genS do > Append(i,[t36,s36]); > od; #In view of Lemma 5.5, every subgroup generated by the generating set obtained above is a subgroup of index dividing $2$ in primitive subgroup of $GL(36,2)$. #So we need to find the normalizer $N=N_{GL(36,2)}(F)$ having the structure $(F.Sp(4,3)).4$. In order to find it we need to generate an element x36 of order 4 #such that x36 normalizes F and $t36^x36=4$. gap> Display(t36); . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . 1 #We see that t36 can be divided into 9\times 9 blocks, each block has structure I_9\otimes (2\times 2)-matrix. We form 4\times 4 matrix out of these 2\times 2 matrices gap> z:=Z(2); Z(2)^0 gap> t4:=[[0*z,0*z,z,0*z],[0*z,0*z,0*z,z],[0*z,z,z,0*z],[z,z,0*z,z]]; [ [ 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2) ], [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ] ] gap> Display(t); . . 1 . . . . 1 . 1 1 . 1 1 . 1 gap> Order(t); 15 #Now we find $x\in GL(4,2)$ such that t^x=t^2, and then construct x36 by using this x, making the reverse procedure with respect to the procedure of obtaining t #out of t36 gap> x:=RepresentativeAction(GL(4,2),t4,t4^2);; gap> x1:=[[x[1][1],x[1][2]],[x[2][1],x[2][2]]];; gap> x2:=[[x[1][3],x[1][4]],[x[2][3],x[2][4]]];; gap> x3:=[[x[3][1],x[3][2]],[x[4][1],x[4][2]]];; gap> x4:=[[x[3][3],x[3][4]],[x[4][3],x[4][4]]];; gap> x118:=KroneckerProduct(I9,x1);; gap> x218:=KroneckerProduct(I9,x2);; gap> x318:=KroneckerProduct(I9,x3);; gap> x418:=KroneckerProduct(I9,x4);; gap> x36:=[]; gap> for i in [1..18] do > Append(x118[i],x218[i]); > x36[i]:=x118[i]; > od; gap> for i in [19..36] do > Append(x318[i-18],x418[i-18]); > x36[i]:=x318[i-18]; > od; gap> Display(x36); . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 gap> t36^x36=t36^2; true gap> x36^2=s36; true #We check, that x36 normalizes the first subgroup from our list, so normalizes $F$ gap> L:=Subgroup(GL(36,2),genS[1]);; gap> L^x36=L; true #We generate the normalizer $N=N_{GL(36,2)}(F)$ of order $(p^a-1)*e*e*|Sp(4,3)|*a=(2^4-1)*9*9*3^4*(3^2-1)*(3^4-1)*4$, we take a subgroup from our list that #is not normalizes by x36 gap> L:=Subgroup(GL(36,2),genS[5]); gap> L^x36=L; false gap> gen:=Concatenation(genS[5],[x36]);; gap> N:=Subgroup(GL(36,2),gen);; gap> n:=Size(N); 251942400 gap> n=(2^4-1)*9*9*3^4*(3^2-1)*(3^4-1)*4; true #Now we run the following loop. Each entry of the list result has three items: the first is the #number of the group in the list genS, the second is the order of the group, and the third is the order of the stabilizer of v in H. #If the order of the stabilizer is 1, then |v^H=|H| and Corollary 2.5 can be applied. gap> v:=[z,0*z,z,z,0*z,z,z,z,0*z,z,z,z,z,0*z,z,z,z,z,z,0*z,z,z,z,z,z,z,0*z,z,z,z,z,z,z,z,0*z,z];; gap> result:=[];; gap> for j in [1..Size(genS)] do > LL:=Subgroup(GL(36,2),genS[j]);; > H:=Normalizer(N,LL); > n:=Size(H); > m:=Size(v^H); > result[j]:=[j,n,n/m]; > Display([j,n,n/m]); > od; [ 1, 5598720, 1 ] [ 2, 5598720, 1 ] [ 3, 5598720, 1 ] [ 4, 2799360, 1 ] [ 5, 194400, 1 ] [ 6, 933120, 1 ] [ 7, 1555200, 1 ]