As abstract polytope cakix is isomorphic to rackix, thereby replacing octagrams by octagons, resp. stop by op and gocco by socco, resp. tistodip by todip, goccope by soccope, and gittith by steth, resp scant by quacant, stotet by otet, tragocco by trasocco, gittithip by stethip, and ginnont by sinnont.

Incidence matrix according to Dynkin symbol

x3o3o3o3x4/3x4*d

. . . . .   .    | 1920 |    4    4    4 |    6   12   12    6    6   4 |    4   12   12   12   12   12    4   4   6 |   1   4   4    6   6  12   4   4  12   1   1   4 |  1  1   4   6  4  1
-----------------+------+----------------+------------------------------+--------------------------------------------+--------------------------------------------------+--------------------
x . . . .   .    |    2 | 3840    *    * |    3    3    3    0    0   0 |    3    6    6    3    3    3    0   0   0 |   1   3   3    3   3   6   1   1   3   0   0   0 |  1  1   3   3  1  0
. . . . x   .    |    2 |    * 3840    * |    0    3    0    3    0   1 |    0    3    0    6    0    3    3   0   3 |   0   1   0    3   0   3   3   0   6   1   0   3 |  1  0   1   3  3  1
. . . . .   x    |    2 |    *    * 3840 |    0    0    3    0    3   1 |    0    0    3    0    6    3    0   3   3 |   0   0   1    0   3   3   0   3   6   0   1   3 |  0  1   1   3  3  1
-----------------+------+----------------+------------------------------+--------------------------------------------+--------------------------------------------------+--------------------
x3o . . .   .    |    3 |    3    0    0 | 3840    *    *    *    *   * |    2    2    2    0    0    0    0   0   0 |   1   2   2    1   1   2   0   0   0   0   0   0 |  1  1   2   1  0  0
x . . . x   .    |    4 |    2    2    0 |    * 5760    *    *    *   * |    0    2    0    2    0    1    0   0   0 |   0   1   0    2   0   2   1   0   2   0   0   0 |  1  0   1   2  1  0
x . . . .   x    |    4 |    2    0    2 |    *    * 5760    *    *   * |    0    0    2    0    2    1    0   0   0 |   0   0   1    0   2   2   0   1   2   0   0   0 |  0  1   1   2  1  0
. . . o3x   .    |    3 |    0    3    0 |    *    *    * 3840    *   * |    0    0    0    2    0    0    2   0   1 |   0   0   0    1   0   0   2   0   2   1   0   2 |  1  0   0   1  2  1
. . . o .   x4*d |    4 |    0    0    4 |    *    *    *    * 2880   * |    0    0    0    0    2    0    0   2   2 |   0   0   0    0   1   0   0   2   2   0   1   2 |  0  1   0   1  2  1
. . . . x4/3x    |    8 |    0    4    4 |    *    *    *    *    * 960 |    0    0    0    0    0    3    0   0   3 |   0   0   0    0   0   3   0   0   6   0   0   3 |  0  0   1   3  3  1
-----------------+------+----------------+------------------------------+--------------------------------------------+--------------------------------------------------+--------------------
x3o3o . .   .    ♦    4 |    6    0    0 |    4    0    0    0    0   0 | 1920    *    *    *    *    *    *   *   * |   1   1   1    0   0   0   0   0   0   0   0   0 |  1  1   1   0  0  0
x3o . . x   .    ♦    6 |    6    3    0 |    2    3    0    0    0   0 |    * 3840    *    *    *    *    *   *   * |   0   1   0    1   0   1   0   0   0   0   0   0 |  1  0   1   1  0  0
x3o . . .   x    ♦    6 |    6    0    3 |    2    0    3    0    0   0 |    *    * 3840    *    *    *    *   *   * |   0   0   1    0   1   1   0   0   0   0   0   0 |  0  1   1   1  0  0
x . . o3x   .    ♦    6 |    3    6    0 |    0    3    0    2    0   0 |    *    *    * 3840    *    *    *   *   * |   0   0   0    1   0   0   1   0   1   0   0   0 |  1  0   0   1  1  0
x . . o .   x4*d ♦    8 |    4    0    8 |    0    0    4    0    2   0 |    *    *    *    * 2880    *    *   *   * |   0   0   0    0   1   0   0   1   1   0   0   0 |  0  1   0   1  1  0
x . . . x4/3x    ♦   16 |    8    8    8 |    0    4    4    0    0   2 |    *    *    *    *    * 1440    *   *   * |   0   0   0    0   0   2   0   0   2   0   0   0 |  0  0   1   2  1  0
. . o3o3x   .    ♦    4 |    0    6    0 |    0    0    0    4    0   0 |    *    *    *    *    *    * 1920   *   * |   0   0   0    0   0   0   1   0   0   1   0   1 |  1  0   0   0  1  1
. . o3o .   x4*d ♦    8 |    0    0   12 |    0    0    0    0    6   0 |    *    *    *    *    *    *    * 960   * |   0   0   0    0   0   0   0   1   0   0   1   1 |  0  1   0   0  1  1
. . . o3x4/3x4*d ♦   24 |    0   24   24 |    0    0    0    8    6   6 |    *    *    *    *    *    *    *   * 480 |   0   0   0    0   0   0   0   0   2   0   0   2 |  0  0   0   1  2  1
-----------------+------+----------------+------------------------------+--------------------------------------------+--------------------------------------------------+--------------------
x3o3o3o .   .    ♦    5 |   10    0    0 |   10    0    0    0    0   0 |    5    0    0    0    0    0    0   0   0 | 384   *   *    *   *   *   *   *   *   *   *   * |  1  1   0   0  0  0
x3o3o . x   .    ♦    8 |   12    4    0 |    8    6    0    0    0   0 |    2    4    0    0    0    0    0   0   0 |   * 960   *    *   *   *   *   *   *   *   *   * |  1  0   1   0  0  0
x3o3o . .   x    ♦    8 |   12    0    4 |    8    0    6    0    0   0 |    2    0    4    0    0    0    0   0   0 |   *   * 960    *   *   *   *   *   *   *   *   * |  0  1   1   0  0  0
x3o . o3x   .    ♦    9 |    9    9    0 |    3    9    0    3    0   0 |    0    3    0    3    0    0    0   0   0 |   *   *   * 1280   *   *   *   *   *   *   *   * |  1  0   0   1  0  0
x3o . o .   x4*d ♦   12 |   12    0   12 |    4    0   12    0    3   0 |    0    0    4    0    3    0    0   0   0 |   *   *   *    * 960   *   *   *   *   *   *   * |  0  1   0   1  0  0
x3o . . x4/3x    ♦   24 |   24   12   12 |    8   12   12    0    0   3 |    0    4    4    0    0    3    0   0   0 |   *   *   *    *   * 960   *   *   *   *   *   * |  0  0   1   1  0  0
x . o3o3x   .    ♦    8 |    4   12    0 |    0    6    0    8    0   0 |    0    0    0    4    0    0    2   0   0 |   *   *   *    *   *   * 960   *   *   *   *   * |  1  0   0   0  1  0
x . o3o .   x4*d ♦   16 |    8    0   24 |    0    0   12    0   12   0 |    0    0    0    0    6    0    0   2   0 |   *   *   *    *   *   *   * 480   *   *   *   * |  0  1   0   0  1  0
x . . o3x4/3x4*d ♦   48 |   24   48   48 |    0   24   24   16   12  12 |    0    0    0    8    6    6    0   0   2 |   *   *   *    *   *   *   *   * 480   *   *   * |  0  0   0   1  1  0
. o3o3o3x   .    ♦    5 |    0   10    0 |    0    0    0   10    0   0 |    0    0    0    0    0    0    5   0   0 |   *   *   *    *   *   *   *   *   * 384   *   * |  1  0   0   0  0  1
. o3o3o .   x4*d ♦   16 |    0    0   32 |    0    0    0    0   24   0 |    0    0    0    0    0    0    0   8   0 |   *   *   *    *   *   *   *   *   *   * 120   * |  0  1   0   0  0  1
. . o3o3x4/3x4*d ♦   64 |    0   96   96 |    0    0    0   64   48  24 |    0    0    0    0    0    0   16   8   8 |   *   *   *    *   *   *   *   *   *   *   * 120 |  0  0   0   0  1  1
-----------------+------+----------------+------------------------------+--------------------------------------------+--------------------------------------------------+--------------------
x3o3o3o3x   .    ♦   30 |   60   60    0 |   60   90    0   60    0   0 |   30   60    0   60    0    0   30   0   0 |   6  15   0   20   0   0  15   0   0   6   0   0 | 64  *   *   *  *  *
x3o3o3o .   x4*d ♦  160 |  320    0  320 |  320    0  480    0  240   0 |  160    0  320    0  240    0    0  80   0 |  32   0  80    0  80   0   0  40   0   0  10   0 |  * 12   *   *  *  *
x3o3o . x4/3x    ♦   32 |   48   16   16 |   32   24   24    0    0   4 |    8   16   16    0    0    6    0   0   0 |   0   4   4    0   0   4   0   0   0   0   0   0 |  *  * 240   *  *  *
x3o . o3x4/3x4*d ♦   72 |   72   72   72 |   24   72   72   24   18  18 |    0   24   24   24   18   18    0   0   3 |   0   0   0    8   6   6   0   0   3   0   0   0 |  *  *   * 160  *  *
x . o3o3x4/3x4*d ♦  128 |   64  192  192 |    0   96   96  128   96  48 |    0    0    0   64   48   24   32  16  16 |   0   0   0    0   0   0  16   8   8   0   0   2 |  *  *   *   * 60  *
. o3o3o3x4/3x4*d ♦  160 |    0  320  320 |    0    0    0  320  240  80 |    0    0    0    0    0    0  160  80  40 |   0   0   0    0   0   0   0   0   0  32  10  10 |  *  *   *   *  * 12