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

xno xno xno   (n>2)

. . . . . .    | nnn |    3 |   3  12 |  12   8 |  3  12 |  6
---------------+-----+------+---------+---------+--------+---
x . . . . .  & |   2 | 3nnn |   1   4 |   6   4 |  2   8 |  5
---------------+-----+------+---------+---------+--------+---
xno . . . .  & |   n |    n | 3nn   * |   4   0 |  2   4 |  4
x . x . . .  & |   4 |    4 |   * 3nn |   2   2 |  1   5 |  4
---------------+-----+------+---------+---------+--------+---
xno x . . .  & ♦  2n |   3n |   2   n | 6nn   * |  1   2 |  3
x . x . x .    ♦   8 |   12 |   0   6 |   * nnn |  0   3 |  3
---------------+-----+------+---------+---------+--------+---
xno xno . .  & ♦  nn |  2nn |  2n  nn |  2n   0 | 3n   * |  2
xno x . x .  & ♦  4n |   8n |   4  5n |   4   n |  * 3nn |  2
---------------+-----+------+---------+---------+--------+---
xno xno x .  & ♦ 2nn |  5nn |  4n 4nn |  6n  nn |  2  2n | 3n

The incidence matrix of the complex polytope thus is

 nnn |   3 |  3   verf = regular real triangle
-----+-----+---
   n | 3nn |  2
-----+-----+---
♦ nn |  2n | 3n

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

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