$Version

HC1:=(({{\[Placeholder], \[Placeholder], d, d, d, d, d, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6}, {\[Placeholder], \[Placeholder], xy, yz, zx, \(x² - y²\), z², x, y, z, x, y, z, x, y, z, x, y, z, x, y, z, x, y, z}, {d, xy, 0, 0, 0, 0, 0, 0, -pdπ1, 0, 0, pdπ2, 0, -pdπ3, 0, 0, pdπ4, 0, 0, 0, 0, 0, 0, 0, 0}, {d, yz, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -pdπ3, 0, 0, pdπ4, 0, pdπ5, 0, 0, -pdπ6, 0}, {d, zx, 0, 0, 0, 0, 0, 0, 0, -pdπ1, 0, 0, pdπ2, 0, 0, 0, 0, 0, 0, pdπ5, 0, 0, -pdπ6, 0, 0}, {d, \(x² - y²\), 0, 0, 0, 0, 0, -(1/2)(Sqrt[3])pdσ1, 0, 0, (1/2) (Sqrt[3]) 
                  pdσ2, 0, 0, 0, (1/2)(Sqrt[3])pdσ3, 0, 0, -(1/2) (Sqrt[3]) pdσ4, 0, 0, 0, 0, 0, 0, 0}, {d, z², 0, 0, 0, 0, 0, (1/2) pdσ1, 0, 0, -(1/2) 
                  pdσ2, 0, 0, 0, (1/2) pdσ3, 0, 0, -(1/2) pdσ4, 0, 0, 0, pdσ5, 0, 0, -pdσ6}, {1, x, 0, 0, 0, -(1/2)(Sqrt[3])pdσ1, (1/2) pdσ1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, y, -pdπ1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {1, z, 0, 0, -pdπ1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {2, x, 0, 0, 0, (1/2) (Sqrt[3]) pdσ2, -(1/2) 
                  pdσ2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {2, y, pdπ2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {2, z, 0, 0, pdπ2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {3, x, -pdπ3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {3, y, 0, 0, 0, (1/2)(Sqrt[3])pdσ3, (1/2) pdσ3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {3, z, 0, -pdπ3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {4, x, pdπ4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {4, y, 0, 0, 0, -(1/2) (Sqrt[3]) pdσ4, -(1/2) 
                  pdσ4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {4, z, 0, pdπ4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {5, x, 0, 0, pdπ5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {5, y, 0, pdπ5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {5, z, 0, 0, 0, 0, pdσ5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {6, x, 0, 0, -pdπ6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {6, y, 0, -pdπ6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {6, z, 0, 0, 0, 0, -pdσ6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}}))[[Range[3, 25], Range[3, 25]]]

HC1C := 
        HC1[[Range[1, 5], 
          Range[6, 23]]];
HC1C2 := 
        HC1[[Range[6, 23], 
          Range[1, 5]]];

HC1C - Transpose[HC1C2] // MatrixForm

Q = QRDecomposition[
          Transpose[HC1C]][[1]]; 
    Length[Q]

L = NullSpace[HC1C]; Length[L]

T = IdentityMatrix[Length[HC1]];

T[[Range[6, 6 + Length[Q] - 1], 
          Range[6, Length[HC1]]]] = Q;

T[[Range[6 + Length[Q], Length[HC1]], 
          Range[6, Length[HC1]]]]  =  
        Table[L[[i]]/(Sqrt[Conjugate[L][[i]] . L[[i]]]), {i, 1, Length[L]}];

T = 
        Assuming[
          pdσ1 < 0 && pdπ1 > 0 && pdσ2 < 0 && pdπ2 > 0 && 
            pdσ3 < 0 && pdπ3 > 0 && pdσ4 < 0 && 
            pdπ4 > 0 && pdσ5 < 0 && pdπ5 > 0 && 
            pdσ6 < 0 && pdπ6 > 0, FullSimplify[T]];

myHC1 = T . HC1 . Transpose[T] // FullSimplify; 
    myHC1 // MatrixForm