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The graph M?C`CHC_aC@HCK}??

This graph is embeddable. It is even 5-grid embeddable.

Embedding

An embedding of this graph is given by.

(0, 4, 2)
(2, -4, 0)
(4, 0, 4)
(2, 4, 4)
(4, -4, 2)
(4, -4, -4)
(0, 4, 0)
(2, -2, 4)
(4, 2, 2)
(4, 0, 0)
(0, 0, 4)
(4, 4, 0)
(0, 4, -4)
(4, 2, -4)