Acronym	...
Name	dual of pentagonal prism, (ex derived) pedpy variant, o3m3o5o cell
	©
Vertex figure	[t5], [T4]
Dihedral angles	at long edge:   arccos[-(11+4 sqrt(5))/41] = 119.107231° at short edge:   arccos[-(11+4 sqrt(5))/41] = 119.107231°
Dual	pentagonal prism
Face vector	7, 15, 10
Confer	more general: m mNo   related Johnson solids: pedpy   general polytopal classes: Catalan polyhedra
External links

The triangles {(t,T,T)} have vertex angles t = acos[(1+5 sqrt(5))/16] = 40.423604° resp. T = acos[(5-sqrt(5))/8] = 69.788198°.

Note that the term pentagonal dipyramid, although stated above, in general says nothing about the relative ratio of the to be chosen lacing edge size. Here the edge ratio is chosen such as to match their duality to the pentagonal prism.

The c edge, provided in the below description, only qualifies as pseudo edge wrt. the full polyhedron.

Incidence matrix according to Dynkin symbol

co oo5ox&#zy   → height = 0
                 c = (5+3 sqrt(5))/5 = 2.341641 (tip-to-tip distance)
                 y = 1+1/sqrt(5) = 1.447214 (lacing edge size)
(tegum product of c-line with {5})

o. o.5o.    | 2 * |  5 0 |  5
.o .o5.o    | * 5 |  2 2 |  4
------------+-----+------+---
oo oo5oo&#y | 1 1 | 10 * |  2
.. .o5.x    | 0 2 |  * 5 |  2
------------+-----+------+---
.. .. ox&#y | 1 2 |  2 1 | 10

oxo3ooo&#yt   → both heights = c/2 = (5+3 sqrt(5))/10 = 1.170820
                y = 1+1/sqrt(5) = 1.447214 (lacing edge size)
(pt || pseudo {5} || pt)

o..5o..    | 1 * * | 5 0 0 | 5 0
.o.5.o.    | * 5 * | 1 2 1 | 2 2
..o5..o    | * * 1 | 0 0 5 | 0 5
-----------+-------+-------+---
oo.5oo.&#y | 1 1 0 | 5 * * | 2 0
.x.5...    | 0 2 0 | * 5 * | 1 1
.oo5.oo&#y | 0 1 1 | * * 5 | 0 2
-----------+-------+-------+----
ox.5...&#y | 1 2 0 | 2 1 0 | 5 *
.xo5...&#y | 0 2 1 | 0 1 2 | * 5