Acronym	cid
Name	small complex icosidodecahedron, (as type C:) complex polyhedron x2-3-o2-5-o2
Circumradius	sqrt[(5+sqrt(5))/8] = 0.951057
Vertex figure	[(3/2,5)5] (type A) or [(3,5)5]/3 (type B)
Snub derivation
General of army	ike
Colonel of regiment	ike
Confer	non-Grünbaumian masters: gad   ike   Grünbaumian relatives: ike+2gad   2ike+gad   ike+3gad   3ike+gad   3ike+3gad   2ike+4gad   4ike+2gad   convex uniform variant: tid   variations: a3b5c   general polytopal classes: Wythoffian polyhedra
External links

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). – Moreover it also could be taken, as Quickfur pointed out in 2022, as a different compound as well: the chiral-tetrahedrally symmetric compound of 4 teddi (type D). Then however icosahedral edges fall into 2 classes of coincident edges: either 2 incident 3-5 edges or an incident pair of a 3-3 and a 5-5 edge.

Through the replacement of normal pentagons by complete pentagons (-x)5f, i.e. additional f-edges, this same set of faces provides an even further type E. Vertices coincide by 5 then, x-edges by pairs.

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

starting figure: x3o5o

(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

as chiral-tetrahedrally symmetrical compound (type D)

 12 |  2  4  2  2 | 1  3  5 1 || 3
----+-------------+-----------++--
  2 | 12  *  *  * | 1  1  0 0 || 1
  2 |  * 24  *  * | 0  1  1 0 || 1
  2 |  *  * 12  * | 0  0  2 0 || 1
  2 |  *  *  * 12 | 0  0  1 1 || 1
----+-------------+-----------++--
  3 |  3  0  0  0 | 4  *  * * || 1
  3 |  1  2  0  0 | * 12  * * || 1
  5 |  0  2  2  1 | *  * 12 * || 1
  3 |  0  0  0  3 | *  *  * 4 || 1
----+-------------+-----------++--
♦ 9 |  3  6  3  3 | 1  3  3 1 || 4

o3(-x)5f (Type E)

.   .  . | 60 |  2  1 |  1  2
---------+----+-------+------
.  -x  . |  2 | 60  * |  1  1
.   .  f |  2 |  * 30 |  0  2
---------+----+-------+------
o3(-x) . |  3 |  3  0 | 20  *
. (-x)5f | 10 |  5  5 |  * 12