Acronym  nappy 
Name  nantiprismatical pyramid 
Segmentochoron display  
Circumradius  sqrt[(22 cos(π/n))/(56 cos(π/n))] 
General of army  (is itself convex) 
Colonel of regiment  (is itself locally convex) 
Especially  pen (n=2)* octpy (n=3) squappy (n=4) pappy (n=5) 
Confer 

* The case n=2 equally would be considerable here by concept, it just has a different incidence matrix as the ngons become degenerate.
Note the conceptual difference between this antiprismpyramid and the pyramidantiprism napt. In fact, the latter happens to be the external blend of 2 of the former.
Solving the height formula below for the limitting (then degenerate) case, results in n ≤ π/arccos(5/6) = 5.363958.
Incidence matrix according to Dynkin symbol
pt  nap → height = sqrt[(56 cos(π/n))/(88 cos(π/n))]
1 * ♦ 2n 0 0  2n 2n 0 0  2 2n 0
* 2n  1 2 2  2 2 1 3  1 3 1
+++
1 1  2n * *  2 2 0 0  1 3 0
0 2  * 2n *  1 0 1 1  1 1 1
0 2  * * 2n  0 1 0 2  0 2 1
+++
1 2  2 1 0  2n * * *  1 1 0
1 2  2 0 1  * 2n * *  0 2 0
0 n  0 n 0  * * 2 *  1 0 1 {n}
0 3  0 1 2  * * * 2n  0 1 1
+++
1 n  n n 0  n 0 1 0  2 * * npy
1 3  3 1 2  1 2 0 1  * 2n * tet
0 2n  0 2n 2n  0 0 2 2n  * * 1 nap
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