(type A)   (type B)

Looks like a compound of 2 cuboctahedra (co), and indeed vertices, edges, and {4} all coincide by pairs. In type B additionally the {3} coincide by pairs.

Incidence matrix according to Dynkin symbol

x3/2x4o4*a (type A)

.   . .    | 24 |  2  2 | 2 1 1
-----------+----+-------+------
x   . .    |  2 | 24  * | 1 1 0
.   x .    |  2 |  * 24 | 1 0 1
-----------+----+-------+------
x3/2x .    |  6 |  3  3 | 8 * *
x   . o4*a |  4 |  4  0 | * 6 *
.   x4o    |  4 |  0  4 | * * 6

x4/3o4/3x3/2*a (type A)

.   .   .      | 24 |  2  2 | 1 2 1
---------------+----+-------+------
x   .   .      |  2 | 24  * | 1 1 0
.   .   x      |  2 |  * 24 | 0 1 1
---------------+----+-------+------
x4/3o   .      |  4 |  4  0 | 6 * *
x   .   x3/2*a |  6 |  3  3 | * 8 *
.   o4/3x      |  4 |  0  4 | * * 6

β3β3x (type B)

both( . . . ) | 24 |  1  2  1 | 1 1  2
--------------+----+----------+-------
both( . . x ) |  2 | 12  *  * | 0 1  1
sefa( s3s . ) |  2 |  * 24  * | 1 0  1
sefa( . β3x ) |  2 |  *  * 12 | 0 1  1
--------------+----+----------+-------
both( s3s . ) ♦  3 |  0  3  0 | 8 *  *
      . β3x   ♦  6 |  3  0  3 | * 4  *
sefa( β3β3x ) |  4 |  1  2  1 | * * 12

starting figure: x3x3x

x3β3x (type A)

both( . . . ) | 24 |  1  1  1  1 | 1 1 1 1
--------------+----+-------------+--------
both( x . . ) |  2 | 12  *  *  * | 1 0 1 0
both( . . x ) |  2 |  * 12  *  * | 0 1 1 0
sefa( x3β . ) |  2 |  *  * 12  * | 1 0 0 1
sefa( . β3x ) |  2 |  *  *  * 12 | 0 1 0 1
--------------+----+-------------+--------
      x3β .   ♦  6 |  3  0  3  0 | 4 * * *
      . β3x   ♦  6 |  0  3  0  3 | * 4 * *
both( x . x ) |  4 |  2  2  0  0 | * * 6 *
sefa( x3β3x ) |  4 |  0  0  2  2 | * * * 6

starting figure: x3x3x

β3x4o (type A)

both( . . . ) | 24 |  2  2 | 2 1 1
--------------+----+-------+------
both( . x . ) |  2 | 24  * | 1 1 0
sefa( β3x . ) |  2 |  * 24 | 1 0 1
--------------+----+-------+------
      β3x .   ♦  6 |  3  3 | 8 * *
both( . x4o ) |  4 |  4  0 | * 6 *
sefa( β3x4o ) |  4 |  0  4 | * * 6

starting figure: x3x4o