Looks like a compound of 2 pentachora (pen) with (in type A) vertices and edges identified, while faces and cells coincide by pairs.

There also is a non-uniform type B too, which occurs as {6/2}-scalene.

Incidence matrix according to Dynkin symbol

x3o3o3o3/2*b   (type A)

. . . .      | 5 ♦  4 | 12 | 4 4
-------------+---+----+----+----
x . . .      | 2 | 10 |  6 | 3 3
-------------+---+----+----+----
x3o . .      | 3 |  3 | 20 | 1 1
-------------+---+----+----+----
x3o3o .      ♦ 4 |  6 |  4 | 5 *
x3o . o3/2*b ♦ 4 |  6 |  4 | * 5

x3o3o3/2o3*b   (type A)

. . .   .    | 5 ♦  4 | 12 | 4 4
-------------+---+----+----+----
x . .   .    | 2 | 10 |  6 | 3 3
-------------+---+----+----+----
x3o .   .    | 3 |  3 | 20 | 1 1
-------------+---+----+----+----
x3o3o   .    ♦ 4 |  6 |  4 | 5 *
x3o .   o3*b ♦ 4 |  6 |  4 | * 5

or
. . .   .      | 5 ♦  4 | 12 |  8
---------------+---+----+----+---
x . .   .      | 2 | 10 |  6 |  6
---------------+---+----+----+---
x3o .   .      | 3 |  3 | 20 |  2
---------------+---+----+----+---
x3o3o   .    & ♦ 4 |  6 |  4 | 10

x3o3/2o3/2o3/2*b   (type A)

. .   .   .      | 5 ♦  4 | 12 | 4 4
-----------------+---+----+----+----
x .   .   .      | 2 | 10 |  6 | 3 3
-----------------+---+----+----+----
x3o   .   .      | 3 |  3 | 20 | 1 1
-----------------+---+----+----+----
x3o3/2o   .      ♦ 4 |  6 |  4 | 5 *
x3o   .   o3/2*b ♦ 4 |  6 |  4 | * 5

or
. .   .   .        | 5 ♦  4 | 12 |  8
-------------------+---+----+----+---
x .   .   .        | 2 | 10 |  6 |  6
-------------------+---+----+----+---
x3o   .   .        | 3 |  3 | 20 |  2
-------------------+---+----+----+---
x3o3/2o   .      & ♦ 4 |  6 |  4 | 10

x3/2o3o3o3/2*b   (type A)

.   . . .      | 5 ♦  4 | 12 | 4 4
---------------+---+----+----+----
x   . . .      | 2 | 10 |  6 | 3 3
---------------+---+----+----+----
x3/2o . .      | 3 |  3 | 20 | 1 1
---------------+---+----+----+----
x3/2o3o .      ♦ 4 |  6 |  4 | 5 *
x3/2o . o3/2*b ♦ 4 |  6 |  4 | * 5

x3/2o3o3/2o3*b   (type A)

.   . .   .    | 5 ♦  4 | 12 | 4 4
---------------+---+----+----+----
x   . .   .    | 2 | 10 |  6 | 3 3
---------------+---+----+----+----
x3/2o .   .    | 3 |  3 | 20 | 1 1
---------------+---+----+----+----
x3/2o3o   .    ♦ 4 |  6 |  4 | 5 *
x3/2o .   o3*b ♦ 4 |  6 |  4 | * 5

or
.   . .   .      | 5 ♦  4 | 12 |  8
-----------------+---+----+----+---
x   . .   .      | 2 | 10 |  6 |  6
-----------------+---+----+----+---
x3/2o .   .      | 3 |  3 | 20 |  2
-----------------+---+----+----+---
x3/2o3o   .    & ♦ 4 |  6 |  4 | 10

x3/2o3/2o3/2o3/2*b   (type A)

.   .   .   .      | 5 ♦  4 | 12 | 4 4
-------------------+---+----+----+----
x   .   .   .      | 2 | 10 |  6 | 3 3
-------------------+---+----+----+----
x3/2o   .   .      | 3 |  3 | 20 | 1 1
-------------------+---+----+----+----
x3/2o3/2o   .      ♦ 4 |  6 |  4 | 5 *
x3/2o   .   o3/2*b ♦ 4 |  6 |  4 | * 5

or
.   .   .   .        | 5 ♦  4 | 12 |  8
---------------------+---+----+----+---
x   .   .   .        | 2 | 10 |  6 |  6
---------------------+---+----+----+---
x3/2o   .   .        | 3 |  3 | 20 |  2
---------------------+---+----+----+---
x3/2o3/2o   .      & ♦ 4 |  6 |  4 | 10

ox6/2oo xo&#x   (type B)   → height = sqrt(5/12) = 0.645497

o.6/2o. o.    | 2 * ♦ 1  6 0 |  6 6 0 | 1 6
.o6/2.o .o    | * 6 ♦ 0  2 2 |  4 1 1 | 2 2
--------------+-----+--------+--------+----
..   .. x.    | 2 0 | 1  * * |  0 6 0 | 0 6
oo6/2oo oo&#x | 1 1 | * 12 * |  2 1 0 | 1 2
.x   .. ..    | 0 2 | *  * 6 |  2 0 1 | 2 1
--------------+-----+--------+--------+----
ox   .. ..&#x | 1 2 | 0  2 1 | 12 * * | 1 1
..   .. xo&#x | 2 1 | 1  2 0 |  * 6 * | 0 2
.x6/2.o ..    | 0 6 | 0  0 6 |  * * 1 | 2 0
--------------+-----+--------+--------+----
ox6/2oo ..&#x ♦ 1 6 | 0  6 6 |  6 0 1 | 2 *  {6/2}-pyramid
ox   .. xo&#x ♦ 2 2 | 1  4 1 |  2 2 0 | * 6

ox3/2ox xo&#x   (type B)   → height = sqrt(5/12) = 0.645497

o.3/2o. o.    | 2 * ♦ 1  6 0 0 | 3 3 6 0 | 1 3 3
.o3/2.o .o    | * 6 ♦ 0  2 1 1 | 2 2 1 1 | 2 1 1
--------------+-----+----------+---------+------
..   .. x.    | 2 0 | 1  * * * | 0 0 6 0 | 0 3 3
oo3/2oo oo&#x | 1 1 | * 12 * * | 1 1 1 0 | 1 1 1
.x   .. ..    | 0 2 | *  * 3 * | 2 0 0 1 | 2 1 0
..   .x ..    | 0 2 | *  * * 3 | 0 2 0 1 | 2 0 1
--------------+-----+----------+---------+------
ox   .. ..&#x | 1 2 | 0  2 1 0 | 6 * * * | 1 1 0
..   ox ..&#x | 1 2 | 0  2 0 1 | * 6 * * | 1 0 1
..   .. xo&#x | 2 1 | 1  2 0 0 | * * 6 * | 0 1 1
.x3/2.x ..    | 0 6 | 0  0 3 3 | * * * 1 | 2 0 0
--------------+-----+----------+---------+------
ox3/2ox ..&#x ♦ 1 6 | 0  6 3 3 | 3 3 0 1 | 2 * *  {6/2}-pyramid
ox   .. xo&#x ♦ 2 2 | 1  4 1 0 | 2 0 2 0 | * 3 *
..   ox xo&#x ♦ 2 2 | 1  4 0 1 | 0 2 2 0 | * * 3