Acronym	esquigybcu, J37 (alt.: pirco)
Name	elongated square gyrobicupola, pseudorhombicuboctahedron, Miller's solid
Circumradius	sqrt[5+2 sqrt(2)]/2 = 1.398966
	© ©
Vertex figures	[3,43]
General of army	(is itself convex)
Colonel of regiment	(is itself locally convex)
Dihedral angles	between {3} and {4}:   arccos[-sqrt(2/3)] = 144.735610° between {4} and {4}:   135°
Face vector	24, 48, 26
Confer	uniform relative: op   sirco   related Johnson solids: squacu   escu   general polytopal classes: Johnson solids
External links

There also is a non-gyrated axial stack too: squacu + op + ortho squacu = sirco, one of the Archimedean solids. In fact, both have the same vertex figure all over. But sirco features full cubical symmetry, i.e. its symmetry acts transiently, while esquigybcu features a 4-fold antiprismatic symmetry only, thereby dividing its vertex set into 2 classes. Thence it can be called pseudouniform at most.

And just as sirco allows for a quasi-version, querco, this gyrated stack too will have an according quasi-version: rasquacu + inv stop + gyro rasquacu = gyquerco.

Incidence matrix according to Dynkin symbol

xxxo4oxxx&#xt   → height(1,2) = height(3,4) = 1/sqrt(2) = 0.707107
                  height(2,3) = 1
({4} || pseudo {8} || pseudo {8} || dual {4})

o...4o...    | 4 * * * | 2 2 0 0 0 0 0 0 0 | 1 2 1 0 0 0 0 0
.o..4.o..    | * 8 * * | 0 1 1 1 1 0 0 0 0 | 0 1 1 1 1 0 0 0
..o.4..o.    | * * 8 * | 0 0 0 0 1 1 1 1 0 | 0 0 0 1 1 1 1 0
...o4...o    | * * * 4 | 0 0 0 0 0 0 0 2 2 | 0 0 0 0 0 1 2 1
-------------+---------+-------------------+----------------
x... ....    | 2 0 0 0 | 4 * * * * * * * * | 1 1 0 0 0 0 0 0
oo..4oo..&#x | 1 1 0 0 | * 8 * * * * * * * | 0 1 1 0 0 0 0 0
.x.. ....    | 0 2 0 0 | * * 4 * * * * * * | 0 1 0 1 0 0 0 0
.... .x..    | 0 2 0 0 | * * * 4 * * * * * | 0 0 1 0 1 0 0 0
.oo.4.oo.&#x | 0 1 1 0 | * * * * 8 * * * * | 0 0 0 1 1 0 0 0
..x. ....    | 0 0 2 0 | * * * * * 4 * * * | 0 0 0 1 0 1 0 0
.... ..x.    | 0 0 2 0 | * * * * * * 4 * * | 0 0 0 0 1 0 1 0
..oo4..oo&#x | 0 0 1 1 | * * * * * * * 8 * | 0 0 0 0 0 1 1 0
.... ...x    | 0 0 0 2 | * * * * * * * * 4 | 0 0 0 0 0 0 1 1
-------------+---------+-------------------+----------------
x...4o...    | 4 0 0 0 | 4 0 0 0 0 0 0 0 0 | 1 * * * * * * *
xx.. ....&#x | 2 2 0 0 | 1 2 1 0 0 0 0 0 0 | * 4 * * * * * *
.... ox..&#x | 1 2 0 0 | 0 2 0 1 0 0 0 0 0 | * * 4 * * * * *
.xx. ....&#x | 0 2 2 0 | 0 0 1 0 2 1 0 0 0 | * * * 4 * * * *
.... .xx.&#x | 0 2 2 0 | 0 0 0 1 2 0 1 0 0 | * * * * 4 * * *
..xo ....&#x | 0 0 2 1 | 0 0 0 0 0 1 0 2 0 | * * * * * 4 * *
.... ..xx&#x | 0 0 2 2 | 0 0 0 0 0 0 1 2 1 | * * * * * * 4 *
...o4...x    | 0 0 0 4 | 0 0 0 0 0 0 0 0 4 | * * * * * * * 1

or
o...4o...    & | 8  * | 2  2 0 0 0 | 1 2 1 0
.o..4.o..    & | * 16 | 0  1 1 1 1 | 0 1 1 2
---------------+------+------------+--------
x... ....    & | 2  0 | 8  * * * * | 1 1 0 0
oo..4oo..&#x & | 1  1 | * 16 * * * | 0 1 1 0
.x.. ....    & | 0  2 | *  * 8 * * | 0 1 0 1
.... .x..    & | 0  2 | *  * * 8 * | 0 0 1 1
.oo.4.oo.&#x   | 0  2 | *  * * * 8 | 0 0 0 2
---------------+------+------------+--------
x...4o...    & | 4  0 | 4  0 0 0 0 | 2 * * *
xx.. ....&#x & | 2  2 | 1  2 1 0 0 | * 8 * *
.... ox..&#x & | 1  2 | 0  2 0 1 0 | * * 8 *
.xx. ....&#x & | 0  4 | 0  0 1 1 2 | * * * 8