Looks like a compound of 4 tetrahemihexahedra (thah), and indeed vertices, edges, {4} all coincide by four, while {3} coincide by pairs with {6/2}.

Incidence matrix according to Dynkin symbol

β3/2β3x

both( .   . . ) | 24 |  1  2  1 | 1 1  2
----------------+----+----------+-------
both( .   . x ) |  2 | 12  *  * | 0 1  1
sefa( s3/2s . ) |  2 |  * 24  * | 1 0  1
sefa( .   β3x ) |  2 |  *  * 12 | 0 1  1
----------------+----+----------+-------
both( s3/2s . ) ♦  3 |  0  3  0 | 8 *  *
      .   β3x   ♦  6 |  3  0  3 | * 4  *
sefa( β3/2β3x ) |  4 |  1  2  1 | * * 12

starting figure: x3/2x3x