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The graph M`_[@@?G_`GGAF@w?

This graph is embeddable. It is even 13-grid embeddable.

Embedding

An embedding of this graph is given by.

(6, -12, 6)
(12, 2, -8)
(6, -12, 12)
(0, 12, 12)
(0, 6, 12)
(12, 0, 0)
(12, 0, -12)
(12, 0, -6)
(8, 0, 12)
(8, 12, 8)
(12, 8, -8)
(6, 12, 12)
(12, -12, 6)
(0, 12, 0)