Acronym	4,2n-dip
Name	square - 2n-gon duoprism
Circumradius	sqrt[1/2+1/(4 sin2(π/2n))]
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex)
Face vector	8n, 16n, 10n+4, 2n+4
Especially	tes (n=2)   shiddip (n=3)   sodip (n=4)   squadedip (n=5)   sitwadip (n=6)
Confer	general duoprisms: 2n,2m-dip   2n,m-dip   4,n-dip   n,m-dip   general polytopal classes: Wythoffian polychora   segmentochora
External links

Although in here n>1 is allowed, snubbing links require n>2 generally. Nonetheless, s4o2s4o well is possible too (it even becomes a regular polychoron), it just has a different incidence structure because of degeneracies.

Incidence matrix according to Dynkin symbol

x4o x2no   (n>1)

. . .  . | 8n |  2  2 |  1  4 1 |  2 2
---------+----+-------+---------+-----
x . .  . |  2 | 8n  * |  1  2 0 |  2 1
. . x  . |  2 |  * 8n |  0  2 1 |  1 2
---------+----+-------+---------+-----
x4o .  . |  4 |  4  0 | 2n  * * |  2 0
x . x  . |  4 |  2  2 |  * 8n * |  1 1
. . x2no | 2n |  0 2n |  *  * 4 |  0 2
---------+----+-------+---------+-----
x4o x  . ♦  8 |  8  4 |  2  4 0 | 2n *
x . x2no ♦ 4n | 2n 4n |  0 2n 2 |  * 4

snubbed forms: s4o2s2no (for n>2)

x4o xnx   (n>1)

. . . . | 8n |  2  1  1 |  1  2  2 1 | 1 1 2
--------+----+----------+------------+------
x . . . |  2 | 8n  *  * |  1  1  1 0 | 1 1 1
. . x . |  2 |  * 4n  * |  0  2  0 1 | 1 0 2
. . . x |  2 |  *  * 4n |  0  0  2 1 | 0 1 2
--------+----+----------+------------+------
x4o . . |  4 |  4  0  0 | 2n  *  * * | 1 1 0
x . x . |  4 |  2  2  0 |  * 4n  * * | 1 0 1
x . . x |  4 |  2  0  2 |  *  * 4n * | 0 1 1
. . xnx | 2n |  0  n  n |  *  *  * 4 | 0 0 2
--------+----+----------+------------+------
x4o x . ♦  8 |  8  4  0 |  2  4  0 0 | n * *
x4o . x ♦  8 |  8  0  4 |  2  0  4 0 | * n *
x . xnx ♦ 4n | 2n 2n 2n |  0  n  n 2 | * * 4

snubbed forms: s4o2snx (for even n>2), s4o2sns (for n>2)

x x x2no   (n>1)

. . .  . | 8n |  1  1  2 |  1  2  2 1 |  2 1 1
---------+----+----------+------------+-------
x . .  . |  2 | 4n  *  * |  1  2  0 0 |  2 1 0
. x .  . |  2 |  * 4n  * |  1  0  2 0 |  2 0 1
. . x  . |  2 |  *  * 8n |  0  1  1 1 |  1 1 1
---------+----+----------+------------+-------
x x .  . |  4 |  2  2  0 | 2n  *  * * |  2 0 0
x . x  . |  4 |  2  0  2 |  * 4n  * * |  1 1 0
. x x  . |  4 |  0  2  2 |  *  * 4n * |  1 0 1
. . x2no | 2n |  0  0 2n |  *  *  * 4 |  0 1 1
---------+----+----------+------------+-------
x x x  . ♦  8 |  4  4  4 |  2  2  2 0 | 2n * *
x . x2no ♦ 4n | 2n  0 4n |  0 2n  0 2 |  * 2 *
. x x2no ♦ 4n |  0 2n 4n |  0  0 2n 2 |  * * 2

x x xnx   (n>1)

. . . . | 8n |  1  1  1  1 |  1  1  1  1  1 1 | 1 1 1 1
--------+----+-------------+------------------+--------
x . . . |  2 | 4n  *  *  * |  1  1  1  0  0 0 | 1 1 1 0
. x . . |  2 |  * 4n  *  * |  1  0  0  1  1 0 | 1 1 0 1
. . x . |  2 |  *  * 4n  * |  0  1  0  1  0 1 | 1 0 1 1
. . . x |  2 |  *  *  * 4n |  0  0  1  0  1 1 | 0 1 1 1
--------+----+-------------+------------------+--------
x x . . |  4 |  2  2  0  0 | 2n  *  *  *  * * | 1 1 0 0
x . x . |  4 |  2  0  2  0 |  * 2n  *  *  * * | 1 0 1 0
x . . x |  4 |  2  0  0  2 |  *  * 2n  *  * * | 0 1 1 0
. x x . |  4 |  0  2  2  0 |  *  *  * 2n  * * | 1 0 0 1
. x . x |  4 |  0  2  0  2 |  *  *  *  * 2n * | 0 1 0 1
. . xnx | 2n |  0  0  n  n |  *  *  *  *  * 4 | 0 0 1 1
--------+----+-------------+------------------+--------
x x x . ♦  8 |  4  4  4  0 |  2  2  0  2  0 0 | n * * *
x x . x ♦  8 |  4  4  0  4 |  2  0  2  0  2 0 | * n * *
x . xnx ♦ 4n | 2n  0 2n 2n |  0  n  n  0  0 2 | * * 2 *
. x xnx ♦ 4n |  0 2n 2n 2n |  0  0  0  n  n 2 | * * * 2

snubbed forms: s2s2snx (for even n>2), s2s2sns (for n>2)

xx xx2noo&#x   (n>1)   → height = 1
({2n}-p || {2n}-p)

o. o.2no.    | 4n  * |  1  2  1  0  0 |  2 1  1  2  0 0 | 1  2 1 0
.o .o2n.o    |  * 4n |  0  0  1  1  2 |  0 0  1  2  2 1 | 0  2 1 1
-------------+-------+----------------+-----------------+---------
x. ..  ..    |  2  0 | 2n  *  *  *  * |  2 0  1  0  0 0 | 1  2 0 0
.. x.  ..    |  2  0 |  * 4n  *  *  * |  1 1  0  1  0 0 | 1  1 1 0
oo oo2noo&#x |  1  1 |  *  * 4n  *  * |  0 0  1  2  0 0 | 0  2 1 0
.x ..  ..    |  0  2 |  *  *  * 2n  * |  0 0  1  0  2 0 | 0  2 0 1
.. .x  ..    |  0  2 |  *  *  *  * 4n |  0 0  0  1  1 1 | 0  1 1 1
-------------+-------+----------------+-----------------+---------
x. x.  ..    |  4  0 |  2  2  0  0  0 | 2n *  *  *  * * | 1  1 0 0
.. x.2no.    | 2n  0 |  0 2n  0  0  0 |  * 2  *  *  * * | 1  0 1 0
xx ..  ..&#x |  2  2 |  1  0  2  1  0 |  * * 2n  *  * * | 0  2 0 0
.. xx  ..&#x |  2  2 |  0  1  2  0  1 |  * *  * 4n  * * | 0  1 1 0
.x .x  ..    |  0  4 |  0  0  0  2  2 |  * *  *  * 2n * | 0  1 0 1
.. .x2n.o    |  0 2n |  0  0  0  0 2n |  * *  *  *  * 2 | 0  0 1 1
-------------+-------+----------------+-----------------+---------
x. x.2no.    ♦ 4n  0 | 2n 4n  0  0  0 | 2n 2  0  0  0 0 | 1  * * *
xx xx  ..&#x ♦  4  4 |  2  2  4  2  2 |  1 0  2  2  1 0 | * 2n * *
.. xx2noo&#x ♦ 2n 2n |  0 2n 2n  0 2n |  0 1  0 2n  0 1 | *  * 2 *
.x .x2n.o    ♦  0 4n |  0  0  0 2n 4n |  0 0  0  0 2n 2 | *  * * 1

xx xxnxx&#x   (n>1)   → height = 1
({2n}-p || {2n}-p)

o. o. o.    | 4n  * |  1  1  1  1  0  0  0 | 1 1 1  1  1  1 0 0 0 | 1 1 1 1 0
.o .o .o    |  * 4n |  0  0  0  1  1  1  1 | 0 0 0  1  1  1 1 1 1 | 0 1 1 1 1
------------+-------+----------------------+----------------------+----------
x. .. ..    |  2  0 | 2n  *  *  *  *  *  * | 1 1 0  1  0  0 0 0 0 | 1 1 1 0 0
.. x. ..    |  2  0 |  * 2n  *  *  *  *  * | 1 0 1  0  1  0 0 0 0 | 1 1 0 1 0
.. .. x.    |  2  0 |  *  * 2n  *  *  *  * | 0 1 1  0  0  1 0 0 0 | 1 0 1 1 0
oo oonoo&#x |  1  1 |  *  *  * 4n  *  *  * | 0 0 0  1  1  1 0 0 0 | 0 1 1 1 0
.x .. ..    |  0  2 |  *  *  *  * 2n  *  * | 0 0 0  1  0  0 1 1 0 | 0 1 1 0 1
.. .x ..    |  0  2 |  *  *  *  *  * 2n  * | 0 0 0  0  1  0 1 0 1 | 0 1 0 1 1
.. .. .x    |  0  2 |  *  *  *  *  *  * 2n | 0 0 0  0  0  1 0 1 1 | 0 0 1 1 1
------------+-------+----------------------+----------------------+----------
x. x. ..    |  4  0 |  2  2  0  0  0  0  0 | n * *  *  *  * * * * | 1 1 0 0 0
x. .. x.    |  4  0 |  2  0  2  0  0  0  0 | * n *  *  *  * * * * | 1 0 1 0 0
.. x.nx.    | 2n  0 |  0  3  3  0  0  0  0 | * * 2  *  *  * * * * | 1 0 0 1 0
xx .. ..&#x |  2  2 |  1  0  0  2  1  0  0 | * * * 2n  *  * * * * | 0 1 1 0 0
.. xx ..&#x |  2  2 |  0  1  0  2  0  1  0 | * * *  * 2n  * * * * | 0 1 0 1 0
.. .. xx&#x |  2  2 |  0  0  1  2  0  0  1 | * * *  *  * 2n * * * | 0 0 1 1 0
.x .x ..    |  0  4 |  0  0  0  0  2  2  0 | * * *  *  *  * n * * | 0 1 0 0 1
.x .. .x    |  0  4 |  0  0  0  0  2  0  2 | * * *  *  *  * * n * | 0 0 1 0 1
.. .xn.x    |  0 2n |  0  0  0  0  0  3  3 | * * *  *  *  * * * 2 | 0 0 0 1 1
------------+-------+----------------------+----------------------+----------
x. x.nx.    ♦ 4n  0 | 2n 2n 2n  0  0  0  0 | n n 2  0  0  0 0 0 0 | 1 * * * *
xx xx ..&#x ♦  4  4 |  2  2  0  4  2  2  0 | 1 0 0  2  2  0 1 0 0 | * n * * *
xx .. xx&#x ♦  4  4 |  2  0  2  4  2  0  2 | 0 1 0  2  0  2 0 1 0 | * * n * *
.. xxnxx&#x ♦ 2n 2n |  0  n  n 2n  0  n  n | 0 0 1  0  3  3 0 0 1 | * * * 2 *
.x .xn.x    ♦  0 4n |  0  0  0  0 2n 2n 2n | 0 0 0  0  0  0 n n 2 | * * * * 1