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The graph L`HIc?@_?V?eKD

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

Embedding

An embedding of this graph is given by.

(8, -4, -12)
(12, 12, 4)
(6, -12, 6)
(12, 12, 12)
(12, 6, 0)
(4, -8, 12)
(12, -12, 0)
(6, 12, 0)
(12, -6, 0)
(12, -12, 12)
(0, 0, 12)
(6, 12, 6)
(12, 0, -12)