Number of nonisomorphic rings
                             of order (n^4)        (n  is a prime power   ).

U= (({{0, A, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, A}, {0, 0, 1, 0}})); V= (({{0, a, B, B c}, {0, b, 0, B d}, {1, c, 0, a}, {0, d , 0, b}})); Id= (({{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0 , 0, 1}}))

Test256[n][{A,B,a,b,c,d}] returnsTrue if :
V.V == B Id (mod n)
V.U == a Id  + b U+ c V + d U.V (mod n)
In such  case, the ring  Zn(A,B,a,b,c,d) has order n^4.

Id=IdentityMatrix[4]; lon=Length;
V[MOD_][{A_,B_,a_,b_,c_,d_}]=V[MOD_][{A,B,a,b,c,d}]=(({{0, a, B, B c}, {0, b, 0, B d}, {1, c, 0, a}, {0, d , 0, b}}));
U[MOD_][{A_,B_,a_,b_,c_,d_}]=U[MOD_][{A,B,a,b,c,d}]=(({{0, A, 0, 0}, {1, 0, 0, 0}, {0, 0, 0, A}, {0, 0, 1, 0}}));
Test256[MOD_][li_]:=  Mod[MatrixPower[V[MOD][li],2]- li[[2]]*Id ,MOD]==0  Id&&Mod[-V[MOD][li].U[MOD][li]+li[[3]]*Id+li[[4]]U[MOD][li]+li[[5]]V[MOD][li]+ li[[6]]U[MOD][li].V[MOD][li] ,MOD]==0 Id

Test256[4][{1,1,1,1,1,1}]

Test256[4][{-1,-1,0,0,0,-1}]

OLLER[n]  returns  the lists (A,B,a,b,c,d) such that Zn(A,B,a,b,c,d) is of order n^4.

OLLER[MOD_]:= OLLER[MOD]=Mod[Flatten[Table[If[Test256[MOD][{aa,bb,a,b,c,d}] ,{aa,bb,a,b,c,d}],{a,0,MOD-1},{b,0,MOD-1},{c,0,MOD-1},{d,0,MOD-1},{aa,0,MOD-1},{bb,0,MOD-1}],5]//Union,MOD]//Rest//Union;

OLLER[3]

Ring[n][{A,B,a,b,c,d}] returns the n^4 elements of the ring Zn(A,B,a,b,c,d).
ZeroDivisors[n][{A,B,a,b,c,d}] returns the ZeroDivisors elements of the ring Zn(A,B,a,b,c,d).

Ring[MOD_][li_]:=Ring[MOD][li]=Union@Flatten[Mod[Table[x0*Id+x1 U[MOD][li]+x2 V[MOD][li]+ x3 U[MOD][li].V[MOD][li],{x0,MOD},{x1,MOD},{x2,MOD},{x3,MOD}],MOD],3]; ZeroDivisors[MOD_][li_]:=ZeroDivisors[MOD][li]=Select[Ring[MOD][li],Mod[Det[#],FactorInteger[MOD][[1,1]]]==0&];

IsomorphicQ[n] [{A,B,a,b,c,d},{R,S,r,s,t,u,v}] returns True if Zn(A,B,a,b,c,d) and Zn(R,S,r,s,t,u,v) are isomorphic  and False otherwise.

count[i_][j_][s_][t_][MOD_][li_]:={aux=Ring[MOD][li]; Sum[If[Mod[t MatrixPower[aux[[k]],s] -i Id-j aux[[k]],MOD]==0 Id,1,0],{k,lon@aux}]}[[1]]
 mira1[MOD_][li_]:= mira1[MOD][li]=Union[Table[count[i][j][s][t][MOD][li], {i,-1,2},{j,-1,2},{s,2,5},{t,1,1}]]

Generedby[MOD_][{X_,Y_}]:=Length@Union@Mod[Flatten[Table[{x,y,z,t}.{Id,X,Y,X.Y},{x,MOD},{y,MOD},{z,MOD},{t,MOD}],3],MOD]; 
BaseQ[MOD_][{A_,B_,a_,b_,c_,d_},{X_,Y_}]:=    Mod[Y.X-a  Id-b X - c Y -d X.Y,MOD]==0 Id && Generedby[MOD][{X,Y}]==MOD^4

huntisomorphism[MOD_][li1_,li2_] :=huntisomorphism[MOD][li1,li2]={si=False;  RR1=Ring[MOD][li2];R1=Select[RR1,Mod[#.#-li1[[1]] Id,MOD]==0#&];R2=Select[RR1,Mod[#.#-li1[[2]] Id,MOD]==0#&];Do[If[BaseQ[MOD][li1,{R1[[i]],R2[[j]]}],si=True;Break[];],{i,lon@R1},{j,lon@R2}] ;si}[[1]]

IsomorphicQ[MOD_][li1_,li2_]:=IsomorphicQ[MOD][li1,li2]=IsomorphicQ[MOD][li2,li1]=Which[!mira1[MOD][li1]==mira1[MOD][li2],False,True,huntisomorphism[MOD][li1,li2]]

IsomorphicQ[4][{1,1,0,0,0,1},{3,3,0,0,0,3}]

IsomorphicQ[4][{1,1,0,0,0,3},{1,3,0,0,0,3}]

Nonisomorphic[n] returns a complete family of non-isomorphic rings of the type Zn(A,B,a,b,c,d).

$RecursionLimit=1111111111111;Clear[Guapo];Guapo[MOD_][quedan_,ya_]:=Guapo[MOD][quedan,ya]=If[quedan=={},ya,{new=True;Do[If[IsomorphicQ[MOD][quedan[[1]],ya[[i]]],aux=Guapo[MOD][Rest@quedan,ya];new=False;Break[]],{i,lon@ya}]; If[new,yu=Join[ya,{quedan[[1]]}];aux=Guapo[MOD][Rest@quedan, Join[ya,{quedan[[1]]}]],aux];aux}[[1]]]

Nonisomorphic[MOD_]:= Nonisomorphic[MOD]=Guapo[MOD][OLLER[MOD],{}]

Nonisomorphic[2]
Nonisomorphic[3]

Nonisomorphic[4]

A342305[n] returns the number of non-isomorphic rings of  the type Zn(A,B,a,b,c,d) and order n^4 (n is a prime power).

A342305[MOD_]:=Length@Nonisomorphic[MOD]

Table[Print[n,"--->",Timing[A342305[n]]];A342305[n],{n,2,5}]

Nonisomorphic[2]={{0,0,0,0,0,0},{0,0,0,0,0,1},{0,0,1,0,0,1}}

Nonisomorphic[3]=
{{0,0,0,0,0,0},{0,0,0,0,0,1},{0,0,0,0,0,2},{0,0,1,0,0,2},{0,1,0,0,0,1},{0,1,0,0,0,2},{0,1,0,1,0,0},{0,2,0,0,0,1},{0,2,0,0,0,2},{1,0,0,0,1,0},{1,1,0,0,0,1},{1,1,1,1,2,0},{1,2,0,0,0,1}}

Nonisomorphic[4]={{0,0,0,0,0,0},{0,0,0,0,0,1},{0,0,0,0,0,2},{0,0,0,0,0,3},{0,0,0,0,2,1},{0,0,0,0,2,3},{0,0,0,2,2,1},{0,0,0,2,2,3},{0,0,1,0,0,3},{0,0,2,0,0,1},{0,0,2,0,0,3},{0,0,2,0,2,1},{0,0,2,0,2,3},{0,0,2,2,2,1},{0,0,2,2,2,3},{0,1,0,0,0,1},{0,1,0,0,0,3},{0,1,0,0,2,1},{0,1,0,1,0,0},{0,1,0,1,0,2},{0,1,0,2,0,1},{0,1,0,2,0,3},{0,1,0,2,2,3},{0,1,2,0,0,1},{0,1,2,0,2,3},{0,1,2,2,0,3},{0,1,2,2,2,1},{0,2,0,0,0,1},{0,2,0,0,0,3},{0,2,0,0,2,1},{0,2,0,0,2,3},{0,2,0,2,0,1},{0,2,0,2,0,3},{0,2,0,2,2,1},{0,2,0,2,2,3},{0,2,2,0,0,1},{0,2,2,0,2,1},{0,2,2,0,2,3},{0,2,2,2,0,1},{0,2,2,2,0,3},{0,3,0,0,0,1},{0,3,0,0,0,3},{0,3,0,2,0,1},{0,3,0,2,0,3},{0,3,2,0,0,1},{0,3,2,0,2,3},{0,3,2,2,0,3},{0,3,2,2,2,1},{1,0,0,0,1,0},{1,0,0,0,1,2},{1,1,0,0,0,1},{1,1,0,0,2,3},{1,1,0,2,2,1},{1,1,0,2,2,3},{1,1,1,1,1,2},{1,1,1,1,3,0},{1,1,2,0,0,1},{1,1,2,0,2,1},{1,1,2,2,2,3},{1,2,0,0,0,1},{1,2,0,0,0,3},{1,2,0,0,2,1},{1,2,0,0,2,3},{1,2,2,0,0,1},{1,2,2,0,2,3},{1,2,2,2,0,1},{1,2,2,2,0,3},{1,2,2,2,2,1},{1,2,2,2,2,3},{1,3,0,0,0,1},{1,3,0,0,2,3},{1,3,0,2,0,3},{1,3,0,2,2,1},{1,3,2,0,2,1},{1,3,2,2,0,1},{1,3,2,2,2,3},{2,2,0,0,0,1},{2,2,0,0,2,1},{2,2,0,0,2,3},{2,2,2,0,0,1},{2,2,2,0,0,3},{2,2,2,0,2,1},{2,2,2,0,2,3},{2,2,2,2,2,1},{2,2,2,2,2,3},{2,3,0,0,0,1},{2,3,0,0,2,1},{2,3,0,2,0,1},{2,3,0,2,0,3},{2,3,2,0,0,1},{2,3,2,0,2,3},{2,3,2,2,0,3},{2,3,2,2,2,1},{3,3,0,0,2,3},{3,3,0,2,2,1},{3,3,0,2,2,3},{3,3,2,0,2,1}}

Nonisomorphic[5]=
{{0,0,0,0,0,0},{0,0,0,0,0,1},{0,0,0,0,0,2},{0,0,0,0,0,4},{0,0,1,0,0,4},{0,1,0,0,0,1},{0,1,0,0,0,4},{0,1,0,1,0,0},{0,2,0,0,0,1},{0,2,0,0,0,4},{1,0,0,0,1,0},{1,1,0,0,0,1},{1,1,1,1,4,0},{1,2,0,0,0,1}}

Nonisomorphic[8]={{0,0,0,0,0,0},{0,0,0,0,0,2},{0,0,0,0,0,4},{0,0,0,0,0,6},{0,1,0,1,0,0},{0,1,0,1,0,4},{0,4,0,2,0,0},{0,4,0,2,0,2},{0,4,0,2,0,4},{0,4,0,2,0,6},{1,0,0,0,1,0},{1,0,0,0,1,4},{1,1,1,1,3,4},{1,1,1,1,7,0},{1,4,2,2,3,0},{1,4,2,2,3,4},{4,0,0,0,2,0},{4,0,0,0,2,2},{4,0,0,0,2,4},{4,0,0,0,2,6},{4,1,2,1,6,0},{4,1,2,1,6,4},{4,4,4,2,2,0},{4,4,4,2,2,2},{4,4,4,2,2,4},{4,4,4,2,2,6},{0,0,0,0,0,1},{0,0,0,0,4,1},{0,0,0,4,4,1},{0,0,4,0,0,1},{0,0,4,0,4,1},{0,0,4,4,4,1},{0,1,0,0,0,1},{0,1,0,0,4,1},{0,1,0,4,0,1},{0,1,0,4,4,1},{0,1,4,0,0,1},{0,1,4,0,4,1},{0,1,4,4,0,1},{0,1,4,4,4,1},{0,2,0,0,0,1},{0,2,0,0,4,1},{0,2,0,4,0,1},{0,2,0,4,4,1},{0,2,4,0,0,1},{0,2,4,0,4,1},{0,2,4,4,0,1},{0,2,4,4,4,1},{0,3,0,0,0,1},{0,3,0,0,4,1},{0,3,0,4,0,1},{0,3,0,4,4,1},{0,3,4,0,0,1},{0,3,4,0,4,1},{0,3,4,4,0,1},{0,3,4,4,4,1},{0,4,0,0,0,1},{0,4,0,0,4,1},{0,4,0,4,0,1},{0,4,0,4,4,1},{0,5,0,0,0,1},{0,5,0,0,4,1},{0,5,0,4,0,1},{0,5,0,4,4,1},{0,5,4,0,0,1},{0,5,4,0,4,1},{0,5,4,4,0,1},{0,5,4,4,4,1},{0,6,0,0,0,1},{0,6,0,0,4,1},{0,6,0,4,0,1},{0,6,0,4,4,1},{0,6,4,0,0,1},{0,6,4,0,4,1},{0,6,4,4,0,1},{0,6,4,4,4,1},{0,7,0,0,0,1},{0,7,0,0,4,1},{0,7,0,4,0,1},{0,7,0,4,4,1},{0,7,4,0,0,1},{0,7,4,0,4,1},{0,7,4,4,0,1},{0,7,4,4,4,1},{1,1,0,0,0,1},{1,1,0,0,4,1},{1,1,0,4,4,1},{1,1,4,0,0,1},{1,1,4,4,4,1},{1,2,0,0,0,1},{1,2,0,0,4,1},{1,2,0,4,0,1},{1,2,0,4,4,1},{1,2,4,0,0,1},{1,2,4,0,4,1},{1,2,4,4,0,1},{1,2,4,4,4,1},{1,3,0,0,0,1},{1,3,0,0,4,1},{1,3,0,4,0,1},{1,3,0,4,4,1},{1,3,4,0,0,1},{1,3,4,4,0,1},{1,3,4,4,4,1},{1,4,0,0,0,1},{1,4,0,0,4,1},{1,4,4,4,0,1},{1,4,4,4,4,1},{1,5,0,0,0,1},{1,5,0,4,4,1},{1,5,4,4,0,1},{1,6,0,0,0,1},{1,6,0,0,4,1},{1,6,4,4,0,1},{1,6,4,4,4,1},{1,7,0,0,0,1},{1,7,0,4,4,1},{1,7,4,4,0,1},{2,2,0,0,0,1},{2,2,0,0,4,1},{2,2,0,4,4,1},{2,2,4,0,0,1},{2,2,4,0,4,1},{2,2,4,4,4,1},{2,3,0,0,0,1},{2,3,0,0,4,1},{2,3,0,4,0,1},{2,3,0,4,4,1},{2,3,4,0,4,1},{2,3,4,4,4,1},{2,4,0,0,0,1},{2,4,0,0,4,1},{2,4,0,4,0,1},{2,4,0,4,4,1},{2,5,0,0,0,1},{2,5,0,0,4,1},{2,5,0,4,0,1},{2,5,0,4,4,1},{2,5,4,0,0,1},{2,5,4,0,4,1},{2,5,4,4,0,1},{2,5,4,4,4,1},{2,6,0,0,0,1},{2,6,0,0,4,1},{2,6,0,4,0,1},{2,6,0,4,4,1},{2,6,4,0,0,1},{2,6,4,0,4,1},{2,6,4,4,0,1},{2,6,4,4,4,1},{2,7,0,0,0,1},{2,7,0,0,4,1},{2,7,0,4,0,1},{2,7,0,4,4,1},{2,7,4,0,4,1},{2,7,4,4,4,1},{3,3,0,0,4,1},{3,3,4,0,0,1},{3,3,4,4,4,1},{3,4,0,0,0,1},{3,4,0,0,4,1},{3,4,4,4,0,1},{3,4,4,4,4,1},{3,5,0,0,0,1},{3,5,0,4,4,1},{3,5,4,0,4,1},{3,5,4,4,0,1},{3,6,0,4,0,1},{3,6,0,4,4,1},{4,4,0,0,0,1},{4,4,0,0,4,1},{4,4,0,4,4,1},{4,5,0,0,0,1},{4,5,0,4,0,1},{4,5,4,0,4,1},{4,5,4,4,4,1},{4,6,0,0,0,1},{4,6,0,0,4,1},{4,6,0,4,0,1},{4,6,0,4,4,1},{4,7,0,0,0,1},{4,7,0,4,0,1},{4,7,4,0,4,1},{4,7,4,4,4,1},{5,5,0,0,4,1},{5,5,4,4,4,1},{5,6,0,0,0,1},{5,6,0,0,4,1},{5,6,4,4,0,1},{5,6,4,4,4,1},{5,7,0,0,4,1},{5,7,0,4,0,1},{5,7,4,4,4,1},{6,6,0,0,0,1},{6,6,0,0,4,1},{6,6,0,4,4,1},{6,6,4,0,0,1},{6,6,4,0,4,1},{6,6,4,4,4,1},{6,7,0,0,4,1},{6,7,0,4,4,1},{0,0,0,0,0,3},{0,0,0,0,4,3},{0,0,0,4,4,3},{0,0,2,0,0,3},{0,0,2,0,4,3},{0,0,2,4,4,3},{0,0,4,0,0,3},{0,0,4,0,4,3},{0,0,4,4,4,3},{0,1,0,0,0,3},{0,1,0,0,4,3},{0,1,0,4,0,3},{0,1,0,4,4,3},{0,1,2,0,0,3},{0,1,2,0,4,3},{0,1,4,0,0,3},{0,1,4,0,4,3},{0,1,4,4,0,3},{0,1,4,4,4,3},{0,2,0,0,0,3},{0,2,0,0,4,3},{0,2,0,4,0,3},{0,2,0,4,4,3},{0,2,2,0,0,3},{0,2,2,0,4,3},{0,2,2,4,0,3},{0,2,4,0,0,3},{0,2,4,0,4,3},{0,2,4,4,0,3},{0,2,4,4,4,3},{0,3,0,0,0,3},{0,3,0,0,4,3},{0,3,0,4,0,3},{0,3,0,4,4,3},{0,3,2,0,0,3},{0,3,4,0,0,3},{0,3,4,0,4,3},{0,3,4,4,0,3},{0,3,4,4,4,3},{0,4,0,0,0,3},{0,4,0,0,4,3},{0,4,0,4,0,3},{0,4,4,4,0,3},{0,5,0,0,0,3},{0,5,0,4,0,3},{0,5,0,4,4,3},{0,5,4,0,0,3},{0,5,4,0,4,3},{0,5,4,4,0,3},{0,5,4,4,4,3},{0,6,0,0,0,3},{0,6,0,0,4,3},{0,6,0,4,0,3},{0,6,4,4,0,3},{0,7,0,0,0,3},{0,7,0,4,0,3},{0,7,0,4,4,3},{0,7,4,0,0,3},{0,7,4,0,4,3},{0,7,4,4,0,3},{0,7,4,4,4,3},{1,1,0,0,0,3},{1,1,2,0,4,3},{1,1,2,4,4,3},{1,2,0,0,0,3},{1,2,0,0,4,3},{1,2,0,4,0,3},{1,2,0,4,4,3},{1,2,2,0,0,3},{1,2,4,4,0,3},{1,3,2,0,4,3},{1,3,2,4,0,3},{1,3,2,4,4,3},{1,5,2,0,4,3},{1,5,2,4,0,3},{1,5,2,4,4,3},{1,7,2,0,4,3},{2,2,0,0,0,3},{2,2,0,0,4,3},{2,2,0,4,4,3},{2,2,2,0,0,3},{2,2,2,0,4,3},{2,2,2,4,4,3},{2,2,4,0,0,3},{2,2,4,0,4,3},{2,2,4,4,4,3},{2,3,0,0,0,3},{2,3,0,4,0,3},{2,3,0,4,4,3},{2,3,2,0,0,3},{2,3,4,0,0,3},{2,3,4,0,4,3},{2,3,4,4,0,3},{2,3,4,4,4,3},{2,4,0,0,0,3},{2,4,0,0,4,3},{2,4,0,4,0,3},{2,4,4,4,0,3},{2,5,0,0,0,3},{2,5,0,0,4,3},{2,5,0,4,0,3},{2,5,0,4,4,3},{2,6,0,4,0,3},{2,6,4,0,0,3},{2,6,4,0,4,3},{2,6,4,4,0,3},{2,7,0,0,0,3},{2,7,4,0,0,3},{3,3,0,0,0,3},{3,3,2,0,4,3},{3,3,2,4,4,3},{3,5,2,0,4,3},{3,5,2,4,0,3},{3,5,2,4,4,3},{3,7,2,0,4,3},{3,7,2,4,0,3},{3,7,2,4,4,3},{4,4,0,0,0,3},{4,4,0,0,4,3},{4,4,0,4,4,3},{4,4,4,4,4,3},{4,6,0,0,0,3},{4,6,0,0,4,3},{4,6,0,4,0,3},{4,6,0,4,4,3},{4,6,4,4,4,3},{5,5,2,0,4,3},{5,5,2,4,4,3},{5,7,2,0,4,3},{6,6,4,0,0,3},{6,6,4,0,4,3},{7,7,2,0,4,3},{7,7,2,4,4,3},{0,0,0,0,0,5},{0,0,0,0,4,5},{0,0,0,4,4,5},{0,0,4,0,0,5},{0,0,4,0,4,5},{0,0,4,4,4,5},{0,1,0,0,0,5},{0,1,0,0,4,5},{0,1,0,4,0,5},{0,1,0,4,4,5},{0,1,4,0,0,5},{0,1,4,0,4,5},{0,1,4,4,0,5},{0,1,4,4,4,5},{0,2,0,0,0,5},{0,2,0,0,4,5},{0,2,0,4,0,5},{0,2,0,4,4,5},{0,2,4,0,0,5},{0,2,4,0,4,5},{0,2,4,4,0,5},{0,2,4,4,4,5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Nonisomorphic[9]={{0,0,0,0,0,0},{0,0,0,0,0,1},{0,0,0,0,0,2},{0,0,0,0,0,3},{0,0,0,0,0,4},{0,0,0,0,0,6},{0,0,0,0,0,8},{0,0,0,0,3,0},{0,0,0,0,3,3},{0,0,0,0,3,6},{0,0,0,3,0,0},{0,0,0,3,0,3},{0,0,0,3,0,6},{0,0,0,3,3,0},{0,0,0,3,3,3},{0,0,0,3,3,6},{0,0,1,0,0,8},{0,0,3,0,0,2},{0,0,3,0,0,8},{0,1,0,0,0,1},{0,1,0,0,0,8},{0,1,0,1,0,0},{0,1,3,1,6,0},{0,2,0,0,0,1},{0,2,0,0,0,8},{0,3,0,0,0,1},{0,3,0,0,0,2},{0,3,0,0,0,4},{0,3,0,0,0,5},{0,3,0,0,0,7},{0,3,0,0,0,8},{0,3,3,0,0,2},{0,3,3,0,0,5},{0,6,0,0,0,1},{0,6,0,0,0,2},{0,6,0,0,0,4},{0,6,0,0,0,5},{0,6,0,0,0,7},{0,6,0,0,0,8},{0,6,3,0,0,2},{0,6,3,0,0,5},{1,0,0,0,1,0},{1,0,3,3,8,0},{1,1,0,0,0,1},{1,1,1,0,0,8},{1,1,1,1,8,0},{1,2,0,0,0,1},{1,3,0,0,0,1},{1,6,0,0,0,1},{2,2,1,0,0,8},{2,3,0,0,0,1},{3,3,0,0,0,1},{3,3,0,0,0,2},{3,3,0,0,0,4},{3,3,0,0,0,8},{3,3,3,0,0,2},{3,6,0,0,0,1},{3,6,0,0,0,2},{3,6,0,0,0,4},{3,6,0,0,0,5},{3,6,0,0,0,7},{3,6,3,0,0,2},{3,6,3,0,0,5},{6,6,0,0,0,1},{6,6,0,0,0,2},{6,6,0,0,0,4},{6,6,3,0,0,2}}

reversible[n] [{A,B,a,b,c,d}] returns True if Zn(A,B,a,b,c,d)  is reversible  and False otherwise.

Clear[reversible]; reversible[MOD_][li_]:=reversible[MOD][li]= {aux=ZeroDivisors[MOD][li];is=True;Do[If[Mod[aux[[i]].aux[[j]],MOD]==0 aux[[j]]&& (!Mod[ aux[[j]].aux[[i]],MOD]==0 Id ),is=False;Break[],0],{i,2,lon[aux]},{j,2,lon[aux]}];is} [[1]];

symmetric[n] [{A,B,a,b,c,d}] returns True if Zn(A,B,a,b,c,d)  is symmetric  and False otherwise.

Clear[symmetric];symmetric[MOD_][li_]:=symmetric[MOD][li]=If[ reversible[MOD][li]==False,False,{aaa=True;aux=ZeroDivisors[MOD][li];Do[If[Mod[aux[[i]].aux[[j]].aux[[k]],MOD]==0 aux[[j]]&& (!Mod[ aux[[j]].aux[[i]].aux[[k]],MOD]==0 IdentityMatrix[4] ),aaa=False;(*Print[i," ",j," ",k]*);Break[]],{i,2,lon[aux]},{j, i,lon[aux]},{k,j,lon[aux]}];aaa} [[1]]];

reversible[4][{3,0,0,0,0,3}]

Select[Nonisomorphic[4],reversible[4][#]&& !symmetric[4][#]&]

reflexive[n] [{A,B,a,b,c,d}] returns True if Zn(A,B,a,b,c,d)  is reflexive  and False otherwise.

reflexive[MOD_][li_]  :=reflexive[MOD][li]=If[reversible[MOD][li]==True,True,{aux=ZeroDivisors[MOD][li];X=U[MOD][li];Y=V[MOD][li];Is=True;Do[If[Mod[ aux[[i]].aux[[j]],MOD]==Mod[ aux[[i]].X.aux[[j]],MOD]==Mod[ aux[[i]].Y.aux[[j]],MOD]==Mod[ aux[[i]].X.Y.aux[[j]],MOD]==0 X&& (!Mod[ aux[[j]].aux[[i]],MOD]==0 X ||!Mod[ aux[[j]].X.aux[[i]],MOD]==0 X || !Mod[ aux[[j]].Y.aux[[i]],MOD]==0 X ||!Mod[ aux[[j]].X.Y.aux[[i]],MOD]==0 X),Is=False;Break[]],{i,2,lon@aux},{j,2,lon@aux}];Is }[[1]]]

semicommutative[n] [{A,B,a,b,c,d}] returns True if Zn(A,B,a,b,c,d)  is semicommutative and False otherwise.

semicommutative[MOD_][li_]:=semicommutative[MOD][li]= If[reversible[MOD][li]==True,True,{aux=ZeroDivisors[MOD][li];X=U[MOD][li];Y=V[MOD][li]; Id=IdentityMatrix[4];aaa=True;Do[If[Mod[aux[[i]].aux[[j]],MOD]==0 aux[[j]]&& (!Mod[ aux[[i]].aux[[j]],MOD]==0 X ||!Mod[ aux[[i]].X.aux[[j]],MOD]==0 X || !Mod[ aux[[i]].Y.aux[[j]],MOD]==0 X ||!Mod[ aux[[i]].X.Y.aux[[j]],MOD]==0 X),aaa=False;Break[],0],{i,lon[aux]},{j,lon[aux]}];aaa} [[1]]];

abelian[n] [{A,B,a,b,c,d}] returns True if Zn(A,B,a,b,c,d)  is abelian and False otherwise.

idempotentQ[MOD_][li_][i_]:={aux=Ring[MOD][li];Mod[aux[[i]].aux[[i]]-aux[[i]],MOD]==0  aux[[i]]}[[1]];
abelian[MOD_][li_]:=abelian[MOD][li]=Sum[If[idempotentQ[MOD][li][i]&&!centerQ[MOD][i][li],1,0],{i,MOD^4}]==0;
centerQ[MOD_][i_][li_]:={aus=Ring[MOD][li];es=True;Do[If[!Mod[aus[[i]].aus[[j]]-aus[[j]].aus[[i]],MOD]==0 Id,es=False;Break[]],{j,1,MOD^4}];es}[[1]]
center[MOD_][li_]:=Table[If[ centerQ[MOD][i][li],Ring[MOD][li][[i]],Id],{i,4}]//Union

Select[Nonisomorphic[4],abelian[4][#]&&reflexive[4][#]&& ! semicommutative[4][#]&]

Select[Nonisomorphic[4],reflexive[4][#]&& ! semicommutative[4][#]&]