Thank you very much. As shown below, it implements the functionality you have described above:

AffModOneDotOnLeft // ClearAll;
AffModOneDotOnLeft[list_List] := Module[{dim,elms},
elms=list;
dim=Dimensions[list[[1]]][[1]]-1;

(*TransformationMatrix[AffineTransform[{#[[1;;dim,1;;dim]],Mod[#[[1;;dim,dim+1]],1]}]&@Dot[x,y]]*)
TransformationMatrix[
    AffineTransform[{#[[1 ;; dim, 1 ;; dim]], 
        Mod[#[[1 ;; dim, dim + 1]], 1]}] &@(Dot@@elms)] // Expand // 
  Together // FullSimplify
  (*{dim,elms}*)
]
AffModOneDotOnLeft[args__] := AffModOneDotOnLeft[{args}]

(*p={{-1, -1, 1, 3/4}, {1, -1, -1, -(5/4)}, {-1, 1, -1, -(5/4)}, {0, 0, 0,
   1}};
q={{-1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}};
r=Inverse[p];

MapAt[Mod[#,1]&, p.q.r,{1;;3,4}]==AffModOneDotOnLeft[p,q.r]==AffModOneDotOnLeft[p,q,r]==AffModOneDotOnLeft[{p,q,r}]*)

In[345]:= AffCrystGroupOnLeft//ClearAll;
AffCrystGroupOnLeft[gens_]:=Module[{dim,count,max,ge},
If[Length[Dimensions[gens]]==3 && Dimensions[gens][[2]]==Dimensions[gens][[3]],
dim=Dimensions[gens][[2]]-1;,Abort[]];

count=0;
max=3;
ge = NestWhile[ (count++;
                Union[Join[#1, 
                    (*TransformationMatrix/@(AffineTransform[{#[[1;;dim,1;;dim]],Mod[#[[1;;dim,dim+1]],1]}]&/@Flatten[Outer[Dot, gens, #1, 1], 1])*)
                    Flatten[Outer[AffModOneDotOnLeft, gens, #1, 1], 1]
                    ]]) &
                ,
                {AffineTransform[{IdentityMatrix[dim],ConstantArray[0, dim]}]//TransformationMatrix}
                ,
                Length[#2] != Length[#1]&
                ,
                2
                ,
                max
            ];

            If[count == max, 
             Failure["InvalidRange",  <|"MessageTemplate" -> "Maximum steps exceeded the limit `Number`","MessageParameters"-> <|"Number" -> max|>|>], 
                  ge]
]

In[347]:= AffCrystGroupOnLeft[SGGenElemAK227LeftS1]

Out[347]= Failure["InvalidRange", <|"MessageTemplate" -> "Maximum \
steps exceeded the limit `Number`", 
  "MessageParameters" -> <|"Number" -> 3|>|>]