Acronym ...
Name qo3oo&#h,
taller variant of trigonal pyramid,
vertex figure of tico
Circumradius sqrt(27/28) = 0.981981
Lace city
in approx. ASCII-art 
 o  
    
q  o
General of army (is itself convex)
Colonel of regiment (is itself locally convex)
Face vector 4, 6, 4
Confer
uniform variant:
tet  
variations:
xo3oo&#q  
uniform relative:
cube  
general pyramid:
n/d-py  

using edge sizes h = sqrt(3) = 1.732051 and q = sqrt(2) = 1.414214


Incidence matrix according to Dynkin symbol

qo3oo&#h   → height = 1
(q-{3} || pt)

o.3o.    | 3 * | 2 1 | 1 2
.o3.o    | * 1 | 0 3 | 0 3
---------+-----+-----+----
q. ..    | 2 0 | 3 * | 1 1
oo3oo&#h | 1 1 | * 3 | 0 2
---------+-----+-----+----
q.3o.    | 3 0 | 3 0 | 1 *  verf(cube)
qo ..&#h | 2 1 | 1 2 | * 3  verf(toe)

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