§4

The order |G| — how many states?

If the cube were fully free (disassemble and reassemble at will), the count would be:

∣ free cube ∣ = 8! \cdot 12! \cdot 3^{8} \cdot 2^{12} = 519, 024, 039, 293, 878, 272, 000

But without disassembly, three independent constraints kick in (§5), each halving the state count:

∣ G ∣ = \frac{8 ! \cdot 12 ! \cdot 3 ^{8} \cdot 2 ^{12}}{3 \cdot 2 \cdot 2}

43,252,003,274,489,856,000

|G| — order of the Rubik's cube group

= 2^{27} \cdot 3^{14} \cdot 5^{3} \cdot 7^{2} \cdot 11

Forty-three quintillion. At one state per second, it would take 1.37 trillion years, dwarfing the age of the universe. At a billion states per second, still 1,370 years.

"You can lay out all cube positions, one millimetre apart, and the line stretches from Earth to the Sun two hundred and fifty-six times over."

— scale of |G|

4.1 Sense of scale

How big is 4.3 × 10¹⁹? Plotted on a logarithmic scale among familiar quantities:

orders of magnitude (log₁₀)

10³

1 thousand

10⁶

1 million

10¹⁰

world population 8 × 10⁹

10¹²

1 trillion

10²⁰

|G| cube states

10²³

stars in observable universe

10²⁵

atoms in a kg of matter ≈ 10²⁵

10²⁶

age of universe (ns) ≈ 10²⁶

4.2 Prime factorization — why group theorists love this number

∣ G ∣ = 2^{27} \cdot 3^{14} \cdot 5^{3} \cdot 7^{2} \cdot 11

This factorization determines the Sylow subgroups of G — one of group theory's sharpest microscopes. For each prime p, the p-Sylow subgroup captures the p-part of |G|:

$P_{2}$ : 2-Sylow, order $2^{27} = 134, 217, 728$ . Contains all half-turn squares
$P_{3}$ : 3-Sylow, order $3^{14} = 4, 782, 969$ . Contains all 3-cycles
$P_{5}$ : 5-Sylow, order $5^{3} = 125$
$P_{7}$ : 7-Sylow, order $7^{2} = 49$
$P_{11}$ : 11-Sylow, order 11

Why does 11 appear?

11 is the largest prime ≤ 12 — it arises from an 11-cycle in S₁₂ (the symmetric group of the 12 edges). Some element of G cycles 11 edges while leaving 1 fixed. Such 11-order elements exist and are concrete witnesses of the prime 11 in |G|.

4.3 Time scales

Rate	Time to enumerate all states	Comparable to
1 / second	1.37 × 10¹² years	100 × age of universe
1 / millisecond	1.37 × 10⁹ years	1/3 the age of Earth
1 / microsecond	1.37 × 10⁶ years	5 × time since Homo sapiens
1 / nanosecond	1370 years	Rome to today
1 / picosecond (10¹²/s)	501 days	—
1 / femtosecond	12 hours	a workday

Even at petaflop scale (10¹⁵ ops/sec), enumerating G outright takes about half a day. This is why the proof of God's number consumed 35 CPU-years and relied on aggressive symmetry reductions — see §11.

4.4 Prime factorisation of |G|

Collecting the product above:

∣ G ∣ = \frac{8 ! \cdot 3 ^{7} \cdot 12 ! \cdot 2 ^{11}}{2} = 2^{27} \cdot 3^{14} \cdot 5^{3} \cdot 7^{2} \cdot 11.

This is an extremely "clean" factorisation — using only the 5 smallest primes, with no 13, 17, 19, etc. The bound comes from the prime factorisations $8! = 2^{7} \cdot 3^{2} \cdot 5 \cdot 7$ and $12! = 2^{10} \cdot 3^{5} \cdot 5^{2} \cdot 7 \cdot 11$ . The 11 is the largest prime factor (since $13 > 12$ ).

227

134,217,728

from 7 + 10 + 11 − 1

314

4,782,969

from 2 + 5 + 7

125

from 1 + 2

from 1 + 1

111

from 0 + 1

Verify: $2^{27} \cdot 3^{14} \cdot 5^{3} \cdot 7^{2} \cdot 11 = 4.325 \times 1 0^{19}$ ✓. This prime structure also constrains §7's attainable element orders (every divisor of |G|) — since 13, 17, 19 don't appear, there is no cube element of order 13 or 17.

4.5 Order comparison across puzzles

puzzle	\|G\|	decimal	vs 3×3
2×2×2 (Pocket)	$\frac{7 ! \cdot 3 ^{6}}{1}$	3,674,160	~10^-13
3×3×3 (this article)	$\frac{8 ! \cdot 3 ^{7} \cdot 12 ! \cdot 2 ^{11}}{2}$	4.33 × 10¹⁹	1.00
4×4×4 (Rubik's Revenge)	$\frac{8 ! \cdot 3 ^{7} \cdot 24 ! ^{2}}{4 ! ^{6} \cdot 24}$	7.40 × 10⁴⁵	1.7 × 10²⁶
5×5×5 (Professor)	$\sim 8! \cdot 3^{7} \cdot 12! \cdot 2^{10} \cdot 24!^{2} \cdot \frac{24 ! ^{2}}{4 ! ^{12}}$	2.83 × 10⁷⁴	6.5 × 10⁵⁴
Megaminx (12 faces)	$\frac{20 ! \cdot 30 ! \cdot 3 ^{19} \cdot 2 ^{29}}{60}$	1.01 × 10⁶⁸	2.3 × 10⁴⁸
Pyraminx (tetrahedron)	$\frac{6 ! \cdot 3 ^{4} \cdot 3 ^{4}}{2}$	75,582,720	~10^-12
Square-1	$\sim 2 \cdot 8! \cdot 8! \cdot 6$	1.55 × 10¹⁰	~10^-10
Skewb	$\frac{8 ! \cdot 3 ^{4}}{12}$	3,149,280	~10^-13

Notable observations: Pyraminx and Pocket are roughly the same order (~10⁷); Square-1 is three orders bigger than Pocket. The 4×4 squares the 3×3 plus extra constants — but the "indistinguishable centres + edge pairs" make computing |G| error-prone (need to divide by 4!⁶ for centres and another 24 for orientation). Megaminx (12 faces) edges out the 5×5 in absolute count, since it has fewer cubies per face than the 5×5.

∎

cuberoot.me · Rubik's Cube as a Group · 2026