207,360 not accounting for rotations, reflections etc.

Dividing by 4 should account for rotations.

Dividing by 4 should account for reflections.

207,360/16 = 12,960 if rotations and reflections are considered the same.

I do not think there are any matrices being counted twice as a reflection and a rotation; I do not think you can make a rotation by doing a single reflection (although I think 2 reflections can be equivalent to a rotation).

The values of these determinants take on 380 different odd integers with the smallest and largest being -407 and +407

-----------

Output:

207360

380

-407 407

-----------

from itertools import permutations

def det3(m):

    x = m[0]*(m[4]*m[8] - (m[5]*m[7]))

    y = m[1]*(m[3]*m[8] - (m[5]*m[6]))

    z = m[2]*(m[3]*m[7] - (m[4]*m[6]))

    return x - y + z

sols = []

integers = []

for perm in permutations([1,2,3,4,5,6,7,8,9]):

    if det3(perm) % 2 == 1:

        sols.append(list(perm))

        integers.append(det3(perm))

integers = sorted(list(set(integers)))

print(len(sols))

print(len(integers))

print(integers[0], integers[-1])