Category:Hexadecachoric group; conjugacy classes

This category is currently an image set, but named like a general category. If images in a different style are added, this category requires diffusion.

See also category: Permutations of tesseract vertices; set partitions.

The hyperoctahedral group of dimension 4 has A000712(4) = 20 conjugacy classes.

names to lists of pairs

'(even, 0) low': [(0, 0)],
'(even, 0) mid': [(3, 0), (5, 0), (6, 0), (9, 0), (10, 0), (12, 0)],
'(even, 0) high': [(15, 0)],

'(even, bold) out': [(0, 7), (0, 16), (0, 23), (3, 7), (5, 16), (6, 23), (9, 23), (10, 16), (12, 7), (15, 7), (15, 16), (15, 23)],
'(even, bold) in': [(3, 16), (3, 23), (5, 7), (5, 23), (6, 7), (6, 16), (9, 7), (9, 16), (10, 7), (10, 23), (12, 16), (12, 23)],

'(even, white) low': [(0, 3), (0, 4), (0, 8), (0, 11), (0, 12), (0, 15), (0, 19), (0, 20), (3, 3), (3, 4), (3, 15), (3, 20), (5, 3), (5, 4), (5, 11), (5, 19), (6, 3), (6, 4), (6, 8), (6, 12), (9, 11), (9, 15), (9, 19), (9, 20), (10, 8), (10, 12), (10, 15), (10, 20), (12, 8), (12, 11), (12, 12), (12, 19)],
'(even, white) high': [(3, 8), (3, 11), (3, 12), (3, 19), (5, 8), (5, 12), (5, 15), (5, 20), (6, 11), (6, 15), (6, 19), (6, 20), (9, 3), (9, 4), (9, 8), (9, 12), (10, 3), (10, 4), (10, 11), (10, 19), (12, 3), (12, 4), (12, 15), (12, 20), (15, 3), (15, 4), (15, 8), (15, 11), (15, 12), (15, 15), (15, 19), (15, 20)],

'(even, green) low': [(0, 1), (0, 2), (0, 5), (0, 6), (0, 14), (0, 21), (3, 1), (5, 5), (6, 2), (9, 21), (10, 14), (12, 6)],
'(even, green) mid': [(3, 2), (3, 5), (3, 14), (3, 21), (5, 1), (5, 2), (5, 6), (5, 21), (6, 1), (6, 5), (6, 6), (6, 14), (9, 1), (9, 5), (9, 6), (9, 14), (10, 1), (10, 2), (10, 6), (10, 21), (12, 2), (12, 5), (12, 14), (12, 21)],
'(even, green) high': [(3, 6), (5, 14), (6, 21), (9, 2), (10, 5), (12, 1), (15, 1), (15, 2), (15, 5), (15, 6), (15, 14), (15, 21)],

'(even, orange)': [(0, 9), (0, 10), (0, 13), (0, 17), (0, 18), (0, 22), (3, 9), (3, 10), (3, 13), (3, 17), (3, 18), (3, 22), (5, 9), (5, 10), (5, 13), (5, 17), (5, 18), (5, 22), (6, 9), (6, 10), (6, 13), (6, 17), (6, 18), (6, 22), (9, 9), (9, 10), (9, 13), (9, 17), (9, 18), (9, 22), (10, 9), (10, 10), (10, 13), (10, 17), (10, 18), (10, 22), (12, 9), (12, 10), (12, 13), (12, 17), (12, 18), (12, 22), (15, 9), (15, 10), (15, 13), (15, 17), (15, 18), (15, 22)],


'(odd, 0) low': [(1, 0), (2, 0), (4, 0), (8, 0)],
'(odd, 0) high': [(7, 0), (11, 0), (13, 0), (14, 0)],

'(odd, bold)': [(1, 7), (1, 16), (1, 23), (2, 7), (2, 16), (2, 23), (4, 7), (4, 16), (4, 23), (7, 7), (7, 16), (7, 23), (8, 7), (8, 16), (8, 23), (11, 7), (11, 16), (11, 23), (13, 7), (13, 16), (13, 23), (14, 7), (14, 16), (14, 23)],

'(odd, white) low': [(1, 3), (1, 4), (1, 11), (1, 15), (1, 19), (1, 20), (2, 3), (2, 4), (2, 8), (2, 12), (2, 15), (2, 20), (4, 3), (4, 4), (4, 8), (4, 11), (4, 12), (4, 19), (7, 3), (7, 4), (8, 8), (8, 11), (8, 12), (8, 15), (8, 19), (8, 20), (11, 15), (11, 20), (13, 11), (13, 19), (14, 8), (14, 12)],
'(odd, white) high': [(1, 8), (1, 12), (2, 11), (2, 19), (4, 15), (4, 20), (7, 8), (7, 11), (7, 12), (7, 15), (7, 19), (7, 20), (8, 3), (8, 4), (11, 3), (11, 4), (11, 8), (11, 11), (11, 12), (11, 19), (13, 3), (13, 4), (13, 8), (13, 12), (13, 15), (13, 20), (14, 3), (14, 4), (14, 11), (14, 15), (14, 19), (14, 20)],

'(odd, green) low': [(1, 1), (1, 5), (1, 21), (2, 1), (2, 2), (2, 14), (4, 2), (4, 5), (4, 6), (8, 6), (8, 14), (8, 21)],
'(odd, green) mid': [(1, 2), (1, 6), (1, 14), (2, 5), (2, 6), (2, 21), (4, 1), (4, 14), (4, 21), (7, 1), (7, 2), (7, 5), (8, 1), (8, 2), (8, 5), (11, 1), (11, 14), (11, 21), (13, 5), (13, 6), (13, 21), (14, 2), (14, 6), (14, 14)],
'(odd, green) high': [(7, 6), (7, 14), (7, 21), (11, 2), (11, 5), (11, 6), (13, 1), (13, 2), (13, 14), (14, 1), (14, 5), (14, 21)],

'(odd, orange)': [(1, 9), (1, 10), (1, 13), (1, 17), (1, 18), (1, 22), (2, 9), (2, 10), (2, 13), (2, 17), (2, 18), (2, 22), (4, 9), (4, 10), (4, 13), (4, 17), (4, 18), (4, 22), (7, 9), (7, 10), (7, 13), (7, 17), (7, 18), (7, 22), (8, 9), (8, 10), (8, 13), (8, 17), (8, 18), (8, 22), (11, 9), (11, 10), (11, 13), (11, 17), (11, 18), (11, 22), (13, 9), (13, 10), (13, 13), (13, 17), (13, 18), (13, 22), (14, 9), (14, 10), (14, 13), (14, 17), (14, 18), (14, 22)],

example in different representations
The following files show the three conjugacy classes low, mid, high in matrix and lattice representation:
matrix	lattice (big permutohedron, small tesseracts)	lattice (big tesseract, small permutohedra)

in matrix
Each of the entries in this 16 × 24 matrix is a permutation of the group. Permutations in the same file with the same color are in the same conjugacy class.

	1 + 6 + 1 = 8	12 + 12 = 24	32 + 32 = 64	12 + 24 + 12 = 48	48
	even (light)			odd (dark)
	4 + 4 = 8	24	32 + 32 = 64	12 + 24 + 12 = 48	48
	odd (dark)			even (light)