Looks like a compound of an octahemioctahedron (oho) with 4 pairs of {3} attached coincidently to a tetrahedral subset of the triangles.

Incidence matrix according to Dynkin symbol

β3/2o3x3*a

both( .   . .    ) | 12 |  2  2  2 | 1 1 2 2
-------------------+----+----------+--------
both( .   . x    ) |  2 | 12  *  * | 1 0 1 0
sefa( β3/2o .    ) |  2 |  * 12  * | 0 1 0 1
sefa( β   . x3*a ) |  2 |  *  * 12 | 0 0 1 1
-------------------+----+----------+--------
both( .   o3x    ) |  3 |  3  0  0 | 4 * * *
      β3/2o .      ♦  3 |  0  3  0 | * 4 * *
      β   . x3*a   ♦  6 |  3  0  3 | * * 4 *
sefa( β3/2o3x3*a ) |  6 |  0  3  3 | * * * 4

starting figure: x3/2o3x3*a

β3/2x3x3*a

demi( .   . .    ) | 12 |  2  2  2 | 2 1 2 1
-------------------+----+----------+--------
both( .   x .    ) |  2 | 12  *  * | 1 1 0 0
both( .   . x    ) |  2 |  * 12  * | 1 0 1 0
sefa( β   . x3*a ) |  2 |  *  * 12 | 0 0 1 1
-------------------+----+----------+--------
both( .   x3x    ) |  6 |  3  3  0 | 4 * * *
      β3/2x .      ♦  3 |  3  0  0 | * 4 * *
      β   . x3*a   ♦  6 |  0  3  3 | * * 4 *
sefa( β3/2x3x3*a ) |  3 |  0  0  3 | * * * 4

starting figure: x3/2x3x3*a