Order-6 truncated triakis tetrahedron (near-miss)

The order-6 truncated triakis tetrahedron is a near-miss Johnson solid. Topologically, its faces are 12 mirror-symmetric pentagons and 4 triambi.

Variations
There are two variants of the order-6 truncated triakis tetrahedron worth mentioning. In both of these variants, the hexagons are regular and the pentagons have two types of internal angles of ~106.77865° and ~112.88538°. The first variant has the unique property of being canonical, if the canonical variant has a midradius of 1, then the two types of edge lengths are $$\frac{6\sqrt2+10\sqrt6}{55}$$ ≈ 0.59964 and $$\frac{4\sqrt6-2\sqrt2}{11}$$ ≈ 0.63359. The second variant is unique for having a circumradius. If the second variant has a circumradius of 1, then the two types of edge lengths are $$\frac{2\sqrt6}{9}$$ ≈ 0.54433 and $$\frac{4\sqrt2}{9}$$ ≈ 0.62854.