Cuboctahedron着色 旋转等价

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用$n$种颜色给Cuboctahedron的面着色 旋转相同视为等价 有多少种方式?

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Octahedral symmetry

Conjugacy classes:

恒同	1个变换	14个不动点
关于4条正三角形轴之一旋转±120°	8个变换	2个不动点
关于3条正方形轴之一旋转±90°	6个变换	2个不动点
关于3条正方形轴之一旋转180°	3个变换	2个不动点
关于6条体对角线旋转180°	6个变换	0个不动点

By Orbit-Stabilizer Theorem and Burnside's lemma, the number of distinct orbits is the average number of points left fixed by an element of $G$.$$|X/G|={\frac {1}{|G|}}\sum _{g\in G}|X^{g}|=\frac{1}{24}\left(1\times n^{14}+8\times n^2+6\times n^2+3\times n^2+6\times n^0\right) = \frac{n^{14}+17n^2+6}{24}$$