I test the following code snippet in GAP:
M:=[[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [1/4, 1/4, 1/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [0, 1/2, 1/2, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [3/4, 1/4, 3/4, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [1/4, 3/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 1/2, 1/2, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [3/4, 3/4, 1/4, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [0, 0, 0, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [0, 0, 0, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [1/2, 1/2, 0, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [1/2, 1/2, 0, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [3/4, 1/4, 3/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [1/4, 1/4, 1/4, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [1/4, 3/4, 3/4, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [1/2, 0, 1/2, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1/4, 1/4, 1/4, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 1/2, 1/2, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [1/4, 3/4, 3/4, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [1/2, 1/2, 0, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [1/4, 1/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [0, 0, 0, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [3/4, 3/4, 1/4, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1/4, 1/4, 1/4, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [3/4, 3/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [1/2, 0, 1/2, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1/2, 1/2, 0, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [3/4, 3/4, 1/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [1/2, 0, 1/2, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1/4, 3/4, 3/4, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [1/2, 1/2, 0, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [3/4, 1/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [1/2, 0, 1/2, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [1/4, 1/4, 1/4, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1/2, 1/2, 0, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [1/2, 1/2, 0, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 0, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [1/4, 3/4, 3/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [3/4, 3/4, 1/4, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [1/2, 1/2, 0, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [3/4, 1/4, 3/4, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [0, 1/2, 1/2, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [3/4, 3/4, 1/4, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1/2, 0, 1/2, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [3/4, 1/4, 3/4, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [3/4, 3/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [1/4, 1/4, 1/4, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [3/4, 3/4, 1/4, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [1/4, 1/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [0, 1/2, 1/2, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1/2, 0, 1/2, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [3/4, 1/4, 3/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [1/2, 1/2, 0, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1/4, 1/4, 1/4, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [1/2, 0, 1/2, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [3/4, 3/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [1/2, 1/2, 0, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [1/4, 3/4, 3/4, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [0, 0, 0, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [0, 0, 0, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1/2, 0, 1/2, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [0, 1/2, 1/2, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [1/2, 0, 1/2, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [0, 1/2, 1/2, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [1/4, 1/4, 1/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [3/4, 1/4, 3/4, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [1/2, 0, 1/2, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [3/4, 3/4, 1/4, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [3/4, 1/4, 3/4, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [1/2, 1/2, 0, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [3/4, 3/4, 1/4, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [0, 1/2, 1/2, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [3/4, 1/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [1/4, 3/4, 3/4, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [3/4, 1/4, 3/4, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [1/4, 3/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [0, 0, 0, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 1/2, 1/2, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [1/4, 3/4, 3/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [3/4, 3/4, 1/4, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [0, 1/2, 1/2, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [1/4, 1/4, 1/4, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [0, 0, 0, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [3/4, 1/4, 3/4, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 1/2, 1/2, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [1/2, 0, 1/2, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1/4, 1/4, 1/4, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [0, 1/2, 1/2, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [3/4, 1/4, 3/4, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [1/2, 0, 1/2, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [3/4, 3/4, 1/4, 1]], [[-1, 0, 0, 0], [0, -1, 0, 0], [0, 0, -1, 0], [1/4, 3/4, 3/4, 1]], [[0, -1, 0, 0], [1, 0, 0, 0], [0, 0, -1, 0], [0, 1/2, 1/2, 1]], [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, -1, 0], [1/4, 1/4, 1/4, 1]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, -1, 0], [1/2, 1/2, 0, 1]], [[-1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1/4, 3/4, 3/4, 1]], [[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]], [[1, 0, 0, 0], [0, -1, 0, 0], [0, 0, 1, 0], [1/4, 1/4, 1/4, 1]], [[0, -1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [1/2, 0, 1/2, 1]], [[0, -1, 0, 0], [0, 0, -1, 0], [-1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, -1, 0], [0, 1, 0, 0], [-1, 0, 0, 0], [0, 1/2, 1/2, 1]], [[0, 1, 0, 0], [0, 0, 1, 0], [-1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, 1, 0], [0, -1, 0, 0], [-1, 0, 0, 0], [1/2, 0, 1/2, 1]], [[0, -1, 0, 0], [0, 0, 1, 0], [1, 0, 0, 0], [1/4, 3/4, 3/4, 1]], [[0, 0, 1, 0], [0, 1, 0, 0], [1, 0, 0, 0], [1/2, 1/2, 0, 1]], [[0, 1, 0, 0], [0, 0, -1, 0], [1, 0, 0, 0], [3/4, 3/4, 1/4, 1]], [[0, 0, -1, 0], [0, -1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1]], [[0, 0, -1, 0], [-1, 0, 0, 0], [0, -1, 0, 0], [1/4, 3/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 0, 1, 0], [0, -1, 0, 0], [0, 1/2, 1/2, 1]], [[0, 0, 1, 0], [1, 0, 0, 0], [0, -1, 0, 0], [3/4, 1/4, 3/4, 1]], [[1, 0, 0, 0], [0, 0, -1, 0], [0, -1, 0, 0], [0, 0, 0, 1]], [[0, 0, -1, 0], [1, 0, 0, 0], [0, 1, 0, 0], [1/4, 3/4, 3/4, 1]], [[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [1/2, 0, 1/2, 1]], [[0, 0, 1, 0], [-1, 0, 0, 0], [0, 1, 0, 0], [3/4, 1/4, 3/4, 1]], [[-1, 0, 0, 0], [0, 0, -1, 0], [0, 1, 0, 0], [1/2, 1/2, 0, 1]]];
g:=AffineCrystGroup(M);
sgs:=SmallGeneratingSet(g);
G:=Group(sgs);
pg1:=PointGroup(g);
pg2:=PointGroup(G);
In the above code snippet, the affine matrix M is obtained from the following cif file:
#(C) 2022 by FIZ Karlsruhe - Leibniz Institute for Information Infrastructure. All rights reserved.
data_136212-ICSD
_database_code_ICSD 136212
_audit_creation_date 2022-02-01
_chemical_name_common Carbon
_chemical_formula_structural C
_chemical_formula_sum C1
_chemical_name_structure_type Diamond#C(cF8)#α−Sn
_exptl_crystal_density_diffrn 3.51
_citation_title
;
Multipole electron densities and structural parameters from synchrotron powder
X-ray diffraction data obtained with a MYTHEN detector system (OHGI)
;
loop_
_citation_id
_citation_journal_full
_citation_year
_citation_journal_volume
_citation_page_first
_citation_page_last
_citation_journal_id_ASTM
primary 'Acta Crystallographica, Section A: Foundations and Advances' 2021 77 85
95 ACSAD7
loop_
_citation_author_citation_id
_citation_author_name
primary 'Svane, Bjarke'
primary 'Tolborg, Kasper'
primary 'Kato, Kenichi'
primary 'Brummerstedt Iversen, Bo'
_cell_length_a 3.567286(4)
_cell_length_b 3.567286(4)
_cell_length_c 3.567286(4)
_cell_angle_alpha 90.
_cell_angle_beta 90.
_cell_angle_gamma 90.
_cell_volume 45.4
_cell_formula_units_Z 8
_space_group_name_H-M_alt 'F d -3 m S'
_space_group_IT_number 227
loop_
_space_group_symop_id
_space_group_symop_operation_xyz
1 'z+1/4, y+1/4, -x+1/4'
2 'y+1/4, x+1/4, -z+1/4'
3 'x+1/4, z+1/4, -y+1/4'
4 'z+1/4, x+1/4, -y+1/4'
5 'y+1/4, z+1/4, -x+1/4'
6 'x+1/4, y+1/4, -z+1/4'
7 'z+1/4, -y+1/4, x+1/4'
8 'y+1/4, -x+1/4, z+1/4'
9 'x+1/4, -z+1/4, y+1/4'
10 'z+1/4, -x+1/4, y+1/4'
11 'y+1/4, -z+1/4, x+1/4'
12 'x+1/4, -y+1/4, z+1/4'
13 '-z+1/4, y+1/4, x+1/4'
14 '-y+1/4, x+1/4, z+1/4'
15 '-x+1/4, z+1/4, y+1/4'
16 '-z+1/4, x+1/4, y+1/4'
17 '-y+1/4, z+1/4, x+1/4'
18 '-x+1/4, y+1/4, z+1/4'
19 '-z+1/4, -y+1/4, -x+1/4'
20 '-y+1/4, -x+1/4, -z+1/4'
21 '-x+1/4, -z+1/4, -y+1/4'
22 '-z+1/4, -x+1/4, -y+1/4'
23 '-y+1/4, -z+1/4, -x+1/4'
24 '-x+1/4, -y+1/4, -z+1/4'
25 '-z, -y, x'
26 '-y, -x, z'
27 '-x, -z, y'
28 '-z, -x, y'
29 '-y, -z, x'
30 '-x, -y, z'
31 '-z, y, -x'
32 '-y, x, -z'
33 '-x, z, -y'
34 '-z, x, -y'
35 '-y, z, -x'
36 '-x, y, -z'
37 'z, -y, -x'
38 'y, -x, -z'
39 'x, -z, -y'
40 'z, -x, -y'
41 'y, -z, -x'
42 'x, -y, -z'
43 'z, y, x'
44 'y, x, z'
45 'x, z, y'
46 'z, x, y'
47 'y, z, x'
48 'x, y, z'
49 'z+1/4, y+3/4, -x+3/4'
50 'y+1/4, x+3/4, -z+3/4'
51 'x+1/4, z+3/4, -y+3/4'
52 'z+1/4, x+3/4, -y+3/4'
53 'y+1/4, z+3/4, -x+3/4'
54 'x+1/4, y+3/4, -z+3/4'
55 'z+1/4, -y+3/4, x+3/4'
56 'y+1/4, -x+3/4, z+3/4'
57 'x+1/4, -z+3/4, y+3/4'
58 'z+1/4, -x+3/4, y+3/4'
59 'y+1/4, -z+3/4, x+3/4'
60 'x+1/4, -y+3/4, z+3/4'
61 '-z+1/4, y+3/4, x+3/4'
62 '-y+1/4, x+3/4, z+3/4'
63 '-x+1/4, z+3/4, y+3/4'
64 '-z+1/4, x+3/4, y+3/4'
65 '-y+1/4, z+3/4, x+3/4'
66 '-x+1/4, y+3/4, z+3/4'
67 '-z+1/4, -y+3/4, -x+3/4'
68 '-y+1/4, -x+3/4, -z+3/4'
69 '-x+1/4, -z+3/4, -y+3/4'
70 '-z+1/4, -x+3/4, -y+3/4'
71 '-y+1/4, -z+3/4, -x+3/4'
72 '-x+1/4, -y+3/4, -z+3/4'
73 '-z, -y+1/2, x+1/2'
74 '-y, -x+1/2, z+1/2'
75 '-x, -z+1/2, y+1/2'
76 '-z, -x+1/2, y+1/2'
77 '-y, -z+1/2, x+1/2'
78 '-x, -y+1/2, z+1/2'
79 '-z, y+1/2, -x+1/2'
80 '-y, x+1/2, -z+1/2'
81 '-x, z+1/2, -y+1/2'
82 '-z, x+1/2, -y+1/2'
83 '-y, z+1/2, -x+1/2'
84 '-x, y+1/2, -z+1/2'
85 'z, -y+1/2, -x+1/2'
86 'y, -x+1/2, -z+1/2'
87 'x, -z+1/2, -y+1/2'
88 'z, -x+1/2, -y+1/2'
89 'y, -z+1/2, -x+1/2'
90 'x, -y+1/2, -z+1/2'
91 'z, y+1/2, x+1/2'
92 'y, x+1/2, z+1/2'
93 'x, z+1/2, y+1/2'
94 'z, x+1/2, y+1/2'
95 'y, z+1/2, x+1/2'
96 'x, y+1/2, z+1/2'
97 'z+3/4, y+1/4, -x+3/4'
98 'y+3/4, x+1/4, -z+3/4'
99 'x+3/4, z+1/4, -y+3/4'
100 'z+3/4, x+1/4, -y+3/4'
101 'y+3/4, z+1/4, -x+3/4'
102 'x+3/4, y+1/4, -z+3/4'
103 'z+3/4, -y+1/4, x+3/4'
104 'y+3/4, -x+1/4, z+3/4'
105 'x+3/4, -z+1/4, y+3/4'
106 'z+3/4, -x+1/4, y+3/4'
107 'y+3/4, -z+1/4, x+3/4'
108 'x+3/4, -y+1/4, z+3/4'
109 '-z+3/4, y+1/4, x+3/4'
110 '-y+3/4, x+1/4, z+3/4'
111 '-x+3/4, z+1/4, y+3/4'
112 '-z+3/4, x+1/4, y+3/4'
113 '-y+3/4, z+1/4, x+3/4'
114 '-x+3/4, y+1/4, z+3/4'
115 '-z+3/4, -y+1/4, -x+3/4'
116 '-y+3/4, -x+1/4, -z+3/4'
117 '-x+3/4, -z+1/4, -y+3/4'
118 '-z+3/4, -x+1/4, -y+3/4'
119 '-y+3/4, -z+1/4, -x+3/4'
120 '-x+3/4, -y+1/4, -z+3/4'
121 '-z+1/2, -y, x+1/2'
122 '-y+1/2, -x, z+1/2'
123 '-x+1/2, -z, y+1/2'
124 '-z+1/2, -x, y+1/2'
125 '-y+1/2, -z, x+1/2'
126 '-x+1/2, -y, z+1/2'
127 '-z+1/2, y, -x+1/2'
128 '-y+1/2, x, -z+1/2'
129 '-x+1/2, z, -y+1/2'
130 '-z+1/2, x, -y+1/2'
131 '-y+1/2, z, -x+1/2'
132 '-x+1/2, y, -z+1/2'
133 'z+1/2, -y, -x+1/2'
134 'y+1/2, -x, -z+1/2'
135 'x+1/2, -z, -y+1/2'
136 'z+1/2, -x, -y+1/2'
137 'y+1/2, -z, -x+1/2'
138 'x+1/2, -y, -z+1/2'
139 'z+1/2, y, x+1/2'
140 'y+1/2, x, z+1/2'
141 'x+1/2, z, y+1/2'
142 'z+1/2, x, y+1/2'
143 'y+1/2, z, x+1/2'
144 'x+1/2, y, z+1/2'
145 'z+3/4, y+3/4, -x+1/4'
146 'y+3/4, x+3/4, -z+1/4'
147 'x+3/4, z+3/4, -y+1/4'
148 'z+3/4, x+3/4, -y+1/4'
149 'y+3/4, z+3/4, -x+1/4'
150 'x+3/4, y+3/4, -z+1/4'
151 'z+3/4, -y+3/4, x+1/4'
152 'y+3/4, -x+3/4, z+1/4'
153 'x+3/4, -z+3/4, y+1/4'
154 'z+3/4, -x+3/4, y+1/4'
155 'y+3/4, -z+3/4, x+1/4'
156 'x+3/4, -y+3/4, z+1/4'
157 '-z+3/4, y+3/4, x+1/4'
158 '-y+3/4, x+3/4, z+1/4'
159 '-x+3/4, z+3/4, y+1/4'
160 '-z+3/4, x+3/4, y+1/4'
161 '-y+3/4, z+3/4, x+1/4'
162 '-x+3/4, y+3/4, z+1/4'
163 '-z+3/4, -y+3/4, -x+1/4'
164 '-y+3/4, -x+3/4, -z+1/4'
165 '-x+3/4, -z+3/4, -y+1/4'
166 '-z+3/4, -x+3/4, -y+1/4'
167 '-y+3/4, -z+3/4, -x+1/4'
168 '-x+3/4, -y+3/4, -z+1/4'
169 '-z+1/2, -y+1/2, x'
170 '-y+1/2, -x+1/2, z'
171 '-x+1/2, -z+1/2, y'
172 '-z+1/2, -x+1/2, y'
173 '-y+1/2, -z+1/2, x'
174 '-x+1/2, -y+1/2, z'
175 '-z+1/2, y+1/2, -x'
176 '-y+1/2, x+1/2, -z'
177 '-x+1/2, z+1/2, -y'
178 '-z+1/2, x+1/2, -y'
179 '-y+1/2, z+1/2, -x'
180 '-x+1/2, y+1/2, -z'
181 'z+1/2, -y+1/2, -x'
182 'y+1/2, -x+1/2, -z'
183 'x+1/2, -z+1/2, -y'
184 'z+1/2, -x+1/2, -y'
185 'y+1/2, -z+1/2, -x'
186 'x+1/2, -y+1/2, -z'
187 'z+1/2, y+1/2, x'
188 'y+1/2, x+1/2, z'
189 'x+1/2, z+1/2, y'
190 'z+1/2, x+1/2, y'
191 'y+1/2, z+1/2, x'
192 'x+1/2, y+1/2, z'
loop_
_atom_type_symbol
_atom_type_oxidation_number
C0+ 0
loop_
_atom_site_label
_atom_site_type_symbol
_atom_site_symmetry_multiplicity
_atom_site_Wyckoff_symbol
_atom_site_fract_x
_atom_site_fract_y
_atom_site_fract_z
_atom_site_U_iso_or_equiv
_atom_site_occupancy
C1 C0+ 8 a 0 0 0 0.01831(16) 1
#End of TTdata_136212-ICSD
The following error will be triggered by the above GAP code snippet:
brk_13> pg1:=PointGroup(g);
<matrix group of size 48 with 47 generators>
brk_13> pg2:=PointGroup(G);
Syntax warning: Unbound global variable in *errin*:2
pg2:=PointGroup(G);
^^^
Error, no method found! For debugging hints type ?Recovery from NoMethodFound
Error, no 2nd choice method found for `IsAffineCrystGroupOnLeftOrRight' on 1 arguments at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
<<compiled GAP code>> from GAPROOT/lib/oper1.g:736 in function PointGroup default method requiring categories and checking properties called from
ErrorNoReturn( no_method_found ); at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
<<compiled GAP code>> from GAPROOT/lib/oper1.g:736 in function PointGroup default method requiring categories and checking properties called from
ErrorNoReturn( no_method_found ); at /home/werner/Public/repo/github.com/gap-system/gap.git/lib/methsel2.g:249 called from
<<compiled GAP code>> from GAPROOT/lib/oper1.g:736 in function PointGroup default method requiring categories and checking properties called from
... at *errin*:2
type 'quit;' to quit to outer loop
As you can see, the PointGroup command doesn't work for a group created with the SmallGeneratingSet of an AffineCrystGroup.
OTOH, the SmallGeneratingSet is rather time-cosuming in this case. Are there any tricks to improve the efficiency?
Any hints for these problems will be highly appreciated.
Update (after Max Horn's answer below):
Dear Max Horn, thank you very much for your trick and explanation. I re-checked your method as follows:
gap> g:=AffineCrystGroup(M);
<matrix group with 192 generators>
gap> iso:=IsomorphismFpGroup(g);;
gap> srcgen := MappingGeneratorsImages(iso)[1];
[ [ [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ -3/4, 1/4, -3/4, 1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, -1, 0, 0 ], [ -1/4, 3/4, -3/4, 1 ] ],
[ [ 0, 1, 0, 0 ], [ 0, 0, -1, 0 ], [ -1, 0, 0, 0 ], [ 1, -1/2, -1/2, 1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, -1, 0 ], [ 0, 0, 0, 1 ] ],
[ [ -1, 0, 0, 0 ], [ 0, -1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1/2, -1/2, 1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 1/2, 0, 1/2, 1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1/2, 1/2, 1 ] ],
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 1, 1 ] ] ]
gap> h := AffineCrystGroup(srcgen);
<matrix group with 8 generators>
gap> Pg:=PointGroup(g);
<matrix group of size 48 with 47 generators>
gap> Ph:=PointGroup(h);
<matrix group of size 48 with 5 generators>
gap> Pg=Ph;
true
Regarding your confusion and inquiry, I provide the following supplements:
- About the "small generating set", my method is based on the following description in this book:
My mistake is that the above method is for finite groups, but here I'm dealing with an infinite group, so it doesn't work.
- About the
brk_13>prompt: it's caused by some of previous abnormally terminated jobs or errors which don't affect the current disussion. Because it takes many keystrokes to jump out of these loops, I didn't do so before I perform the test discussed here.
Regards, HZ
The second group $G$ that you generate from a bunch of matrices is created as a plain matrix group, not a crystallographic group. As such, it knows nothing about things specific to crystallographic groups, such as the "point group".
You don't say why you even want a "small generating set", and you didn't ask about this, but I'll mention anyway that in this particular case the following trick gives a fairly small generating set:
PS: your prompt is
brk_13>which indicates you are 13 levels down in "break loops". I recommend reading the section in the GAP reference manual on break loops.