Acronym	icoap (alt.: icoalico)
Name	icositetrachoral antiprism, icositetrachoron atop dual icositetrachoron
Circumradius	sqrt[3+sqrt(2)]/2 = 1.050501
Lace city in approx. ASCII-art	o c Q c o -- x3o4o3o (ico) O C O -- o3o4o3x (dual ico) where: o = o3o4o (point) c = o3o4x (cube) O = x3o4o (oct) Q = q3o4o (q-oct) C = o3x4o (co)
Dihedral angles (at margins)	at oct between ico and octpy:   arccos[-(sqrt(2)-1)] = 114.469801° ...
Face vector	48, 336, 768, 720, 242
Confer	related segmentotera: tes || ico   related CRFs: rect-icoap   trunc-icoap   ambification: oxo3xoo4oox3oxo&#xt   general polytopal classes: segmentotera   fundamental lace prisms   scaliform
External links

It should be emphasized that not only all vertices are equivalent, but also all faces are regular triangles. Thence this polyteron allows for some of the same operations as regulars do: its truncation results in trunc-icoap and its rectification results in rect-icoap.

Incidence matrix according to Dynkin symbol

xo3oo4oo3ox&#x   → height = sqrt[sqrt(2)-1] = 0.643594
(ico || dual ico)

o.3o.4o.3o.    | 24  * ♦  8   6  0 | 12  24  12  0 |  6  24  24   8  0 | 1  6 12  8  1 0
.o3.o4.o3.o    |  * 24 ♦  0   6  8 |  0  12  24 12 |  0   8  24  24  6 | 0  1  8 12  6 1
---------------+-------+-----------+---------------+-------------------+----------------
x. .. .. ..    |  2  0 | 96   *  * |  3   3   0  0 |  3   6   3   0  0 | 1  3  3  1  0 0
oo3oo4oo3oo&#x |  1  1 |  * 144  * ♦  0   4   4  0 |  0   4   8   4  0 | 0  1  4  4  1 0
.. .. .. .x    |  0  2 |  *   * 96 |  0   0   3  3 |  0   0   3   6  3 | 0  0  1  3  3 1
---------------+-------+-----------+---------------+-------------------+----------------
x.3o. .. ..    |  3  0 |  3   0  0 | 96   *   *  * |  2   2   0   0  0 | 1  2  1  0  0 0
xo .. .. ..&#x |  2  1 |  1   2  0 |  * 288   *  * |  0   2   2   0  0 | 0  1  2  1  0 0
.. .. .. ox&#x |  1  2 |  0   2  1 |  *   * 288  * |  0   0   2   2  0 | 0  0  1  2  1 0
.. .. .o3.x    |  0  3 |  0   0  3 |  *   *   * 96 |  0   0   0   2  2 | 0  0  0  1  2 1
---------------+-------+-----------+---------------+-------------------+----------------
x.3o.4o. ..    ♦  6  0 | 12   0  0 |  8   0   0  0 | 24   *   *   *  * | 1  1  0  0  0 0
xo3oo .. ..&#x ♦  3  1 |  3   3  0 |  1   3   0  0 |  * 192   *   *  * | 0  1  1  0  0 0
xo .. .. ox&#x ♦  2  2 |  1   4  1 |  0   2   2  0 |  *   * 288   *  * | 0  0  1  1  0 0
.. .. oo3ox&#x ♦  1  3 |  0   3  3 |  0   0   3  1 |  *   *   * 192  0 | 0  0  0  1  1 0
.. .o4.o3.x    ♦  0  6 |  0   0 12 |  0   0   0  8 |  *   *   *   * 24 | 0  0  0  0  1 1
---------------+-------+-----------+---------------+-------------------+----------------
x.3o.4o.3o.    ♦ 24  0 | 96   0  0 | 96   0   0  0 | 24   0   0   0  0 | 1  *  *  *  * *
xo3oo4oo ..&#x ♦  6  1 | 12   6  0 |  8  12   0  0 |  1   8   0   0  0 | * 24  *  *  * *
xo3oo .. ox&#x ♦  3  2 |  3   6  1 |  1   6   3  0 |  0   2   3   0  0 | *  * 96  *  * *
xo .. oo3ox&#x ♦  2  3 |  1   6  3 |  0   3   6  1 |  0   0   3   2  0 | *  *  * 96  * *
.. oo4oo3ox&#x ♦  1  6 |  0   6 12 |  0   0  12  8 |  0   0   0   8  1 | *  *  *  * 24 *
.o3.o4.o3.x    ♦  0 24 |  0   0 96 |  0   0   0 96 |  0   0   0   0 24 | *  *  *  *  * 1

or
o.3o.4o.3o.     & | 48 ♦   8   6 |  12  36 |  6  32  24 | 1  7  20
------------------+----+---------+---------+------------+---------
x. .. .. ..     & |  2 | 192   * |   3   3 |  3   6   3 | 1  3   4
oo3oo4oo3oo&#x    |  2 |   * 144 ♦   0   8 |  0   8   8 | 0  2   8
------------------+----+---------+---------+------------+---------
x.3o. .. ..     & |  3 |   3   0 | 192   * |  2   2   0 | 1  2   1
xo .. .. ..&#x  & |  3 |   1   2 |   * 576 |  0   2   2 | 0  1   3
------------------+----+---------+---------+------------+---------
x.3o.4o. ..     & ♦  6 |  12   0 |   8   0 | 48   *   * | 1  1   0
xo3oo .. ..&#x  & ♦  4 |   3   3 |   1   3 |  * 384   * | 0  1   1
xo .. .. ox&#x    ♦  4 |   2   4 |   0   4 |  *   * 288 | 0  0   2
------------------+----+---------+---------+------------+---------
x.3o.4o.3o.     & ♦ 24 |  96   0 |  96   0 | 24   0   0 | 2  *   *
xo3oo4oo ..&#x  & ♦  7 |  12   6 |   8  12 |  1   8   0 | * 48   *
xo3oo .. ox&#x  & ♦  5 |   4   6 |   1   9 |  0   2   3 | *  * 192