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How to generate a matrix group?


I have three 7×7 matrices (with real entries, lots of zeros) and I'd like to check if they generate a finite group (or, more precisely, if the group they generate is of precise order). Would it be possible to do with Mathematica?




These are the matrices. All of order 3. I expect this group to be of order 6048 or 12096.


gMatrix = 
  SparseArray[{{1, 3} -> 1, {2, 1} -> 1, {3, 2} -> 
     1, {4, 4} -> -(1/2), {4, 5} -> -(1/2), {4, 6} -> -(1/2), {4, 
      7} -> -(1/2), {5, 4} -> 1/2, {5, 5} -> -(1/2), {5, 6} -> 
     1/2, {5, 7} -> -(1/2), {6, 4} -> 
     1/2, {6, 5} -> -(1/2), {6, 6} -> -(1/2), {6, 7} -> 1/2, {7, 4} ->
      1/2, {7, 5} -> 
     1/2, {7, 6} -> -(1/2), {7, 7} -> -(1/2), {_, _} -> 0}, 7];


hMatrix = 
  SparseArray[{{1, 5} -> 
     1, {2, 2} -> -(1/2), {2, 3} -> -(1/2), {2, 6} -> 1/2, {2, 7} -> 
     1/2, {3, 2} -> 1/2, {3, 3} -> -(1/2), {3, 6} -> -(1/2), {3, 7} ->
      1/2, {4, 1} -> 1, {5, 4} -> 1, {6, 2} -> -(1/2), {6, 3} -> 
     1/2, {6, 6} -> -(1/2), {6, 7} -> 
     1/2, {7, 2} -> -(1/2), {7, 3} -> -(1/2), {7, 6} -> -(1/2), {7, 
      7} -> -(1/2), {_, _} -> 0}, 7];

kMatrix = 

  SparseArray[{{1, 4} -> -1, {2, 2} -> -(1/2), {2, 3} -> -(1/2), {2, 
      6} -> -(1/2), {2, 7} -> 1/2, {3, 2} -> 
     1/2, {3, 3} -> -(1/2), {3, 6} -> -(1/2), {3, 7} -> -(1/2), {4, 
      5} -> -1, {5, 1} -> 1, {6, 2} -> 1/2, {6, 3} -> 
     1/2, {6, 6} -> -(1/2), {6, 7} -> 1/2, {7, 2} -> -(1/2), {7, 3} ->
      1/2, {7, 6} -> -(1/2), {7, 7} -> -(1/2), {_, _} -> 0}, 7];


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