I have been trying to compute the projective order of the matrix M (written below) over residual class ring Z/ (169) Z but my results show it to be 4826796, in contrast to actual order which is 28392. Could someone please suggest something as how to obtain order over Z / (13)^{2} Z ?
a = matrix ([[20,101,52,52,166,148,46,135,96,51,73,49,128], [166, 164, 159, 66, 123, 50, 144, 85, 29, 116, 22, 93, 10],[158, 152, 90, 65, 20, 167, 27, 96, 109, 154, 127, 164, 76],[120, 154, 132, 110, 22, 113, 115, 51, 25, 104, 108, 82, 33],[43, 148, 131, 45, 81, 2, 164, 145, 117, 157, 4, 108, 61],[134, 23, 151, 120, 151, 44, 30, 1, 76, 32, 60, 132, 165],[121, 40, 83, 4, 56, 88, 3, 134, 100, 85, 88, 18, 3],[23, 20, 20, 31, 66, 24, 41, 126, 47, 137, 33, 112, 49], [143, 18, 44, 26, 89, 109, 118, 148, 35, 16, 35, 122, 150], [144, 51, 47, 143, 109, 164, 52, 38, 92, 50, 98, 60, 104],[70, 165, 89, 80, 28, 75, 19, 110, 101, 41, 155, 78, 67],[123, 147, 54, 4, 60, 133, 49, 151, 30, 32, 157, 108, 82], [85, 139, 50, 70, 124, 168, 87, 63, 13, 104, 58, 107, 113]]);
b= mat.identity(13); c= 73*b; d= zero_matrix(13, 13); M= block_matrix([[ a, c], [b, c]]);
