Acronym	sladid (old: sit)
Name	small lanceal disdodecahedron, strombic icositetrahedron, deltoidal icositetrahedron, tetragonal icosikaitetrahedron
	©
Inradius	sqrt[(78+47 sqrt(2))/68] = 1.457577
Vertex figure	[kLL4], [kLS4], [KSS3]
Coordinates	(a, 0, 0)/sqrt(2)   & all permutations and all changes of sign (kLL vertices, vertex-inscribed a-oct, a = sqrt[5+1/sqrt(2)] = 2.388955) (b, b, 0)/sqrt(2)   & all permutations and all changes of sign (kLS vertices, vertex-inscribed b-co, b = sqrt[10+sqrt(2)]/2 = 1.689246) (c, c, c)/2            & all permutations and all changes of sign (KSS vertices, vertex-inscribed c-cube, c = sqrt[2+sqrt(2)] = 1.847759)
Dihedral angles	at long edge:   arccos[-(7+4 sqrt(2))/17] = 138.117959° at short edge:   arccos[-(7+4 sqrt(2))/17] = 138.117959°
Dual	sirco
Face vector	26, 48, 24
Confer	general polytopal classes: Catalan polyhedra
External links

The kites {(k_LS,k_LL,k_LS,K_SS)} have vertex angles k = k_LS = k_LL = arccos[(2-sqrt(2))/4] = 81.578942° resp. K = K_SS = arccos[-(2+sqrt(2))/8] = 115.263174°.

Edge sizes used here are k_LSK_SS = x = 1 (short) resp. k_LLk_LS = d = 2-1/sqrt(2) = 1.292893 (long). – The pseudo edge of size e is the symmetric diagonal of the kites, while the pseudo edge of size b is the asymmetric diagonal of the kites.

The following description is based on the fact that it also is the convex hull (or tegum sum) of 3 polyhedra, an a-scaled oct (k_LL vertices), a b-scaled co (k_LS vertices), and a c-scaled cube (K_SS vertices). Note moreover that, just as its dual's (sirco) square faces form sets of lacing faces of perfect inscribed ops, here as well the k_LL and k_LS vertices together form 3 mutually orthogonal perfect regular octagons. These octagons will have a circumradius a/q = b and a side length d = k_LLk_LS.

Incidence matrix according to Dynkin symbol

m3o4m =
aoo3obo4ooc&#z(d,x,e)   → height = 0
                          a = kLLkLS x(8,2) = sqrt[5+1/sqrt(2)] = 2.388955
                          b = kLSkLS = a/q = sqrt[10+sqrt(2)]/2 = 1.689246
                          c = kLSKSS x(8,2) = sqrt[2+sqrt(2)] = 1.847759
                          d = kLLkLS = lacing(1,2) = 2-1/sqrt(2) = 1.292893
                          e = kLLKSS = lacing(1,3) = sqrt[3-1/sqrt(2)] = 1.514230

o..3o..4o..         | 6  * * |  4  0 |  4  [kLL4]
.o.3.o.4.o.         | * 12 * |  2  2 |  4  [kLS4]
..o3..o4..o         | *  * 8 |  0  3 |  3  [KSS3]
--------------------+--------+-------+---
oo.3oo.4oo.&#d      | 1  1 0 | 24  * |  2  d
.oo3.oo4.oo&#x      | 0  1 1 |  * 24 |  2  x
--------------------+--------+-------+---
... obo ...&#(d,x)t | 1  2 1 |  2  2 | 24  {(kLS,kLL,kLS,KSS)}