main.py

from functools import lru_cache

def mul_by_a(x):
    return ((x[0] + 1) % 8, x[1], x[2], )


def mul_by_b(x):
    return (x[0] * 3 % 8, (x[1] + 1) % 4, x[2], )


def pow_ab(n):
    res = (0, 0, 0)
    n %= 3
    for i in range(n):
        res = mul_by_a(mul_by_b(res))
    return res


def mul_by_c(x):
    return ((x[0] * 7) % 8, (x[1] * 3) % 4, (x[2] + 1) % 2, )


@lru_cache(None)
def mul(x, y):
    for i in range(x[2]):
        y = mul_by_c(y)
    for i in range(x[1]):
        y = mul_by_b(y)
    for i in range(x[0]):
        y = mul_by_a(y)
    return y


def check_assoc(elements):
    elements = sorted(elements)
    for x in elements:
        for y in elements:
            for z in elements:
                u = mul(x, mul(y, z))
                v = mul(mul(x, y), z)
                assert u == v, (x, y, z, u, v)


def check_unity(elements):
    u = (0, 0, 0)
    for x in elements:
        assert mul(x, u) == x, x
        assert mul(u, x) == x, x


def order(x):
    result = 1
    tmp = x
    while tmp != (0, 0, 0):
        tmp = mul(tmp, x)
        result += 1
    return result


def is_gen(s):
    to_check = [(0, 0, 0)]
    checked = set(to_check)
    while to_check:
        x = to_check.pop()
        for e in s:
            p = mul(e, x)
            if p not in checked:
                to_check.append(p)
                checked.add(p)
    return len(checked) == 64


def commute(x, y):
    return mul(x, y) == mul(y, x)


elements = [(i, j, k) for i in range(8) for j in range(4) for k in range(2)]
assert len(elements) == 64


check_assoc(elements)
check_unity(elements)


for e1 in elements:
    for e2 in elements:
        if is_gen([e1, e2]):
            print(e1, e2)
            exit()


for e1 in elements:
    for e2 in elements:
        if commute(e1, e2):
            for e3 in elements:
                if is_gen([e1, e2, e3]):
                    print(e1, e2, e3)
                    exit()