There is a permutation that makes this matrix 2x2 block triangular,
Mmm, I can not find this permutation ...
I found the permutation that makes the matrix block-triangular by eyeball,
based on the Kronecker-product-like spyplot of the matrix:
[ ... ]
This makes it block lower triangular where the first block is 24x24 and
the second 4x4.
well, so long, for the 2x2 block triangularity then ....