Order of an element

For any $g\in G$, there is a smallest positive integer $n$ with $g^n=e$. This $n$ is the order of $g$. Repeat the same alg until you come home — that count is its order.

Some famous orders: R has order 4 (obvious), but R U has order 105 (remarkable), and R U R' U' (the "sexy move") has order 6.

7.1 All attained orders

An element's order in G must divide |G| = 2²⁷ · 3¹⁴ · 5³ · 7² · 11. That allows (27+1)(14+1)(3+1)(2+1)(1+1) = 10,080 divisors. But only 73 are actually attained by elements of G:

Which divisors are missed? For instance $|G|$ itself ($4.3\times 10^{19}$) cannot be an element's order — that would force a cyclic subgroup equal to G, but G is non-Abelian. Most large divisors are similarly out of reach.

7.2 Why 1260 is the maximum

To find a maximal-order element: arrange the corner part into a set of disjoint cycles, and the edge part likewise, maximising the LCM of their lengths. Specifically:

• Corners: a 5-cycle with internal CO summing to 1 mod 3 — local period 5 × 3 = 15
• Edges: a 7-cycle × a 5-cycle of edges with EO summing to 1 mod 2 — local period lcm(7, 4·2) ...
• Total = LCM of the local periods = 2² · 3² · 5 · 7 = 1260

This maximum was first established by analogy with Landau's function g(n) (max order in S_n). Here g(12) = 60, but with the extra CO/EO structure on cubies the cube's maximum bumps up to 1260.

7.3 Landau's function & comparison with Sₙ

The maximum element order in the symmetric group $S_n$ is given by Landau's function $g(n)$: over all partitions $n={\lambda }_1+{\lambda }_2+\cdots$, maximise $\mathrm{lcm}({\lambda }_1,{\lambda }_2,\dots )$. Why: an element of $S_n$ is determined by its disjoint cycle type, and its order equals the lcm of cycle lengths.

The corner sector ($S_8⋉(\mathbb{Z}/3{)}^7$) maxes at $3\cdot g(8)=3\cdot 15=45$, but the cube's "Σco ≡ 0 mod 3" constraint forces orders into the form $\mathrm{lcm}(\text{corner cycles})\cdot 3$. The edge sector maxes at $2\cdot g(12)/k$ (k depends on the parity of EO). Combining both under the parity-coupling constraint yields the maximum 1260.

7.4 All 73 attained orders

The following 73 integers are all attained element orders in G, sorted ascending. Every entry divides $|G|=2^{27}\cdot 3^{14}\cdot 5^3\cdot 7^2\cdot 11$; divisors that do not appear (e.g. 4096) are ruled out by the CO/EO conservation laws.

Pattern: small orders (1–12) appear almost without gaps; 13, 16, 17, 19, 23, 25, 26, 27, 29… are all missing (primes 13, 17, 19, 23 don't divide |G|; 16 and 25 are blocked by CO/EO conservation). Large orders concentrate at products of the form $2^a\cdot 3^b\cdot 5\cdot 7$, peaking at 1260 = 2² · 3² · 5 · 7.