Acronym	...
Name	Shephard's p-generalised cube, complex polyhedron xp-4-o2-3-o2, γp3
	© p=3 p=4 p=5
Circumradius	sqrt(3)/(2 sin(π/p))
Vertex figure	trig
Coordinates	(εpn, εpm, εpk)/(2 sin(π/p))   for any 1≤n,m,k≤p, where εp=exp(2πi/p)
Dual	x2-3-o2-4-op
Face vector	p3, 3p2, 3p
Especially	x3-4-o2-3-o2 (p=3)   x4-4-o2-3-o2 (p=4)   x5-4-o2-3-o2 (p=5)
Confer	more general: xp   xr   xt         xp   xr-4-o2   real space embedding: (p,p,p)-triprism general polytopal classes: complex polytopes
External links

The (complex) edges then are x_p-4-o₂, which were nothing but the set of p-gons of the respective (p,p)-duoprism each. The vertex figure here throughout is just x₂-3-o₂, i.e. nothing but the real space triangle.

Those polytopes happen to be the (complex) 3-dimensional versions of Shephard's generalised hypercubes. In fact, p=2 not only returns into the real subspace only, but moreover becomes (here) nothing but the well-known cube.

The below various incidence representations are direct, next dimensional consequences from what was explained already at x_p-4-o₂ = x_p   x_p. I.e. the according cartesian or prism product applies for complex polytopes alike.

Incidence matrix according to Dynkin symbol

xp-4-o2-3-o2

.    .    .  | p3 |  3  |  3
------------+----+-----+---
xp   .    .  | p  | 3p2 |  2
------------+----+-----+---
xp-4-o2   .  ♦ p2 | 2p  | 3p

snubbed forms: sp-4-o2-3-o2

xp   xp-4-o2

.    .    .  | p3 | 1   2  |  2 1
------------+----+--------+-----
xp   .    .  | p  | p2  *  |  2 0
.    xp   .  | p  | *  2p2 |  1 1
------------+----+--------+-----
xp   xp   .  ♦ p2 | p   p  | 2p *
.    xp-4-o2 ♦ p2 | 0  2p  |  * p

xp   xp   xp

.    .    .  | p3 | 1  1  1  | 1 1 1
------------+----+----------+------
xp   .    .  | p  | p2 *  *  | 1 1 0
.    xp   .  | p  | *  p2 *  | 1 0 1
.    .    xp | p  | *  *  p2 | 0 1 1
------------+----+----------+------
xp   xp   .  ♦ p2 | p  p  0  | p * *
xp   .    xp ♦ p2 | p  0  p  | * p *
.    xp   xp ♦ p2 | 0  p  p  | * * p