Looks like a compound of an octahedra (oct) plus 3 diametral pairs of squares, and indeed edges and {4} coincide by pairs, but vertices are identified (type A). Alternatively it can be seen as a pair of tetrahemihexahedra (thah), mutually in inverted positioning, with vertices pairwise identified.

Incidence matrix according to Dynkin symbol

x3/2o4o4*a (type A)

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

o4/3x3o4*a (type A)

.   . . | 6 |  8 | 4 4
--------+---+----+----
.   x . | 2 | 24 | 1 1
--------+---+----+----
o4/3x . | 4 |  4 | 6 *
.   x3o | 3 |  3 | * 8

o4/3o3x4*a (type A)

.   . .    | 6 |  8 | 4 4
-----------+---+----+----
.   . x    | 2 | 24 | 1 1
-----------+---+----+----
.   o3x    | 3 |  3 | 8 *
o   . x4*a | 4 |  4 | * 6

x4/3o4/3o3/2*a (type A)

.   .   .      | 6 |  8 | 4 4
---------------+---+----+----
x   .   .      | 2 | 24 | 1 1
---------------+---+----+----
x4/3o   .      | 4 |  4 | 6 *
x   .   o3/2*a | 3 |  3 | * 8

β3o4o (type A)

both( . . . ) | 6 |  8 | 4 4
--------------+---+----+----
sefa( β3o . ) | 2 | 24 | 1 1
--------------+---+----+----
      β3o .   ♦ 3 |  3 | 8 *
sefa( β3o4o ) | 4 |  4 | * 6

starting figure: x3o4o

β3/2o4o (type A)

both( .   . . ) | 6 |  8 | 4 4
----------------+---+----+----
sefa( β3/2o . ) | 2 | 24 | 1 1
----------------+---+----+----
      β3/2o .   ♦ 3 |  3 | 8 *
sefa( β3/2o4o ) | 4 |  4 | * 6

starting figure: x3/2o4o

o4/3o3β (type A)

both( .   . . ) | 6 |  8 | 4 4
----------------+---+----+----
sefa( .   o3β ) | 2 | 24 | 1 1
----------------+---+----+----
      .   o3β   ♦ 3 |  3 | 8 *
sefa( o4/3o3β ) | 4 |  4 | * 6

starting figure: o4/3o3x

o4/3o3/2β (type A)

both( .   .   . ) | 6 |  8 | 4 4
------------------+---+----+----
sefa( .   o3/2β ) | 2 | 24 | 1 1
------------------+---+----+----
      .   o3/2β   ♦ 3 |  3 | 8 *
sefa( o4/3o3/2β ) | 4 |  4 | * 6

starting figure: o4/3o3/2x

o3β3o (type A)

both( . . . ) | 6 |  4  4 | 2 2 4
--------------+---+-------+------
sefa( o3β . ) | 2 | 12  * | 1 0 1
sefa( . β3o ) | 2 |  * 12 | 0 1 1
--------------+---+-------+------
      o3β .   ♦ 3 |  3  0 | 4 * *
      . β3o   ♦ 3 |  0  3 | * 4 *
sefa( o3β3o ) | 4 |  2  2 | * * 6

starting figure: o3x3o

o3/2β3o (type A)

both( .   . . ) | 6 |  4  4 | 2 2 4
----------------+---+-------+------
sefa( o3/2β . ) | 2 | 12  * | 1 0 1
sefa( .   β3o ) | 2 |  * 12 | 0 1 1
----------------+---+-------+------
      o3/2β .   ♦ 3 |  3  0 | 4 * *
      .   β3o   ♦ 3 |  0  3 | * 4 *
sefa( o3/2β3o ) | 4 |  2  2 | * * 6

starting figure: o3/2x3o

o3/2β3/2o (type A)

both( .   .   . ) | 6 |  4  4 | 2 2 4
------------------+---+-------+------
sefa( o3/2β   . ) | 2 | 12  * | 1 0 1
sefa( .   β3/2o ) | 2 |  * 12 | 0 1 1
------------------+---+-------+------
      o3/2β   .   ♦ 3 |  3  0 | 4 * *
      .   β3/2o   ♦ 3 |  0  3 | * 4 *
sefa( o3/2β3/2o ) | 4 |  2  2 | * * 6

starting figure: o3/2x3/2o