As abstract polytope cid is isomorphic to gacid, thereby replacing pentagons by pentagrams.

Looks like a compound of the icosahedron (ike) and the great dodecahedron (gad), and indeed edges coincide by pairs, but vertices are identified. Note that without edge-doubling it would be a tetradic figure (type C).

Incidence matrix according to Dynkin symbol

x3/2o5o5*a (type A)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
x   . .    |  2 | 60 |  1  1
-----------+----+----+------
x3/2o .    |  3 |  3 | 20  *
x   . o5*a |  5 |  5 |  * 12

o5/3o3x5*a (type B)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x5*a |  5 |  5 |  * 12

x3/2o5/2o5*a (type B)

.   .   .    | 12 | 10 |  5  5
-------------+----+----+------
x   .   .    |  2 | 60 |  1  1
-------------+----+----+------
x3/2o   .    |  3 |  3 | 20  *
x   .   o5*a |  5 |  5 |  * 12

o5/4x3o5*a (type A)

.   . . | 12 | 10 |  5  5
--------+----+----+------
.   x . |  2 | 60 |  1  1
--------+----+----+------
o5/4x . |  5 |  5 | 12  *
.   x3o |  3 |  3 |  * 20

o5/4o3x5*a (type A)

.   . .    | 12 | 10 |  5  5
-----------+----+----+------
.   . x    |  2 | 60 |  1  1
-----------+----+----+------
.   o3x    |  3 |  3 | 20  *
o   . x5*a |  5 |  5 |  * 12

x5/4o5/2o3*a (type B)

.   .   .    | 12 | 10 |  5  5
-------------+----+----+------
x   .   .    |  2 | 60 |  1  1
-------------+----+----+------
x5/4o   .    |  5 |  5 | 12  *
x   .   o3*a |  3 |  3 |  * 20

o5/4x3/2o5/3*a (type B)

.   .   . | 12 | 10 |  5  5
----------+----+----+------
.   x   . |  2 | 60 |  1  1
----------+----+----+------
o5/4x   . |  5 |  5 | 12  *
.   x3/2o |  3 |  3 |  * 20

x5/4o5/4o3/2*a (type A)

.   .   .      | 12 | 10 |  5  5
---------------+----+----+------
x   .   .      |  2 | 60 |  1  1
---------------+----+----+------
x5/4o   .      |  5 |  5 | 12  *
x   .   o3/2*a |  3 |  3 |  * 20

β3o5o (type A)

both( . . . ) | 12 | 10 |  5  5
--------------+----+----+------
sefa( β3o . ) |  2 | 60 |  1  1
--------------+----+----+------
β3o .     3 |  3 | 20  *
sefa( β3o5o ) |  5 |  5 |  * 12

(Type C)

12 |  5 |  5  5
---+----+------
2 | 30 |  2  2  :4 incident faces
---+----+------
3 |  3 | 20  *
5 |  5 |  * 12

as uniform compound

12 |  5  5 |  5  5 || 1 1
-----+-------+-------++----
2 | 30  * |  2  0 || 1 0
2 |  * 30 |  0  2 || 0 1
-----+-------+-------++----
3 |  3  0 | 20  * || 1 0
5 |  0  5 |  * 12 || 0 1
-----+-------+-------++----
12 | 30  0 | 20  0 || 1 *
12 |  0 30 |  0 20 || * 1