The incidence matrix of the real encasing polytope (n,n,n,n-quip) is

xno xno xno xno   (n>2)

. . . . . . . .    | nnnn |     8 |    4    24 |    24    32 |   6    48   16 |   24   32 |  4  24 |  8
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
x . . . . . . .  & |    2 | 4nnnn |    1     6 |     9    12 |   3    24    8 |   15   20 |  3  18 |  7
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno . . . . . .  & |    n |     n | 4nnn     * |     6     0 |   3    12    0 |   12    8 |  3  12 |  6
x . x . . . . .  & |    4 |     4 |    * 6nnnn |     2     4 |   1    10    4 |    8   12 |  2  13 |  6
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno x . . . . .  & ♦   2n |    3n |    2     n | 12nnn     * |   1     4    0 |    6    4 |  2   8 |  5
x . x . x . . .  & ♦    8 |    12 |    0     6 |     * 4nnnn |   0     3    2 |    3    7 |  1   9 |  5
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno xno . . . .  & ♦   nn |   2nn |   2n    nn |    2n     0 | 6nn     *    * |    4    0 |  2   4 |  4
xno x . x . . .  & ♦   4n |    8n |    4    5n |     4     n |   * 12nnn    * |    2    2 |  1   5 |  4
x . x . x . x .    ♦   16 |    32 |    0    24 |     0     8 |   *     * nnnn |    0    4 |  0   6 |  4
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno xno x . . .  & ♦  2nn |   5nn |   4n   4nn |    6n    nn |   2    2n    0 | 12nn    * |  1   2 |  3
xno x . x . x .  & ♦   8n |   20n |    8   18n |    12    7n |   0     6    n |    * 4nnn |  0   3 |  3
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno xno xno . .  & ♦  nnn |  3nnn |  3nn  3nnn |   6nn   nnn |  3n   3nn    0 |   3n    0 | 4n   * |  2
xno xno x . x .  & ♦  4nn |  12nn |   8n  13nn |   16n   6nn |   4   10n   nn |    4   2n |  * 6nn |  2
-------------------+------+-------+------------+-------------+----------------+-----------+--------+---
xno xno xno x .  & ♦ 2nnn |  7nnn |  6nn  9nnn |  15nn  5nnn |  6n  12nn  nnn |   9n  3nn |  2  3n | 4n

The incidence matrix of the complex polytope thus is

 nnnn ♦    4 |   6 |  4
------+------+-----+---
    n | 4nnn |   3 |  3
------+------+-----+---
♦  nn |   2n | 6nn |  2
------+------+-----+---
♦ nnn |  3nn |  3n | 4n

Generators

with e  = exp(2πi/n),   E  = exp(2πi k/n), 
where 1 < k < n and k not divisor of n

     / e 0 0 0 \         / 0 1 0 0 \         / 1 0 0 0 \         / 1 0 0 0 \
R0 = | 0 E 0 0 | ,  R1 = | 1 0 0 0 | ,  R2 = | 0 0 1 0 | ,  R3 = | 0 1 0 0 |
     | 0 0 E 0 |         | 0 0 1 0 |         | 0 1 0 0 |         | 0 0 0 1 |
     \ 0 0 0 E /         \ 0 0 0 1 /         \ 0 0 0 1 /         \ 0 0 1 0 /

R0n  =  1                                  (rotation-rotation*-rotation*-rotation*)
R12  =  1                                  (exchange 1 & 2)
R22  =  1                                  (exchange 2 & 3)
R22  =  1                                  (exchange 3 & 4)
R0 * R1 * R0 * R1  =  R1 * R0 * R1 * R0    (rot. → up → rot.* → down  =  up → rot.* → down → rot.)
R0 * R2  =  R2 * R0
R0 * R3  =  R3 * R0
R1 * R2 * R1  =  R2 * R1 * R2              (exchange 1 & 3)
R1 * R3  =  R3 * R1
R2 * R3 * R2  =  R3 * R2 * R3              (exchange 2 & 4)