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The {(hhH)²} face has corner angles of h = 108° and H = 144°. Note that h is the same corner angle as the pentagon, thence that non-regular hexagon reuses 4 out of 5 vertices of such a virtually inscribed pentagon.

Despite of being derived from the doe, the total combinatorics of this polyhedron turns out to be the same as that of toe; in fact, as such it is just a toe variant!

Incidence matrix according to Dynkin symbol

xfxx3xxfx&#xt →   all heights = sqrt[(3+sqrt(5))/6] = 0.934172

o...3o...     | 6 * * * | 1 1 1 0 0 0 0 0 0 | 1 1 1 0 0 0
.o..3.o..     | * 6 * * | 0 0 1 1 1 0 0 0 0 | 0 1 1 1 0 0
..o.3..o.     | * * 6 * | 0 0 0 0 1 1 1 0 0 | 0 1 0 1 1 0
...o3...o     | * * * 6 | 0 0 0 0 0 0 1 1 1 | 0 0 0 1 1 1
--------------+---------+-------------------+------------
x... ....     | 2 0 0 0 | 3 * * * * * * * * | 1 1 0 0 0 0
.... x...     | 2 0 0 0 | * 3 * * * * * * * | 1 0 1 0 0 0
oo..3oo..&#x  | 1 1 0 0 | * * 6 * * * * * * | 0 1 1 0 0 0
.... .x..     | 0 2 0 0 | * * * 3 * * * * * | 0 0 1 1 0 0
.oo.3.oo.&#x  | 0 1 1 0 | * * * * 6 * * * * | 0 1 0 1 0 0
..x. ....     | 0 0 2 0 | * * * * * 3 * * * | 0 1 0 0 1 0
..oo3..oo&#x  | 0 0 1 1 | * * * * * * 6 * * | 0 0 0 1 1 0
...x ....     | 0 0 0 2 | * * * * * * * 3 * | 0 0 0 0 1 1
.... ...x     | 0 0 0 2 | * * * * * * * * 3 | 0 0 0 1 0 1
--------------+---------+-------------------+------------
x...3x...     | 6 0 0 0 | 3 3 0 0 0 0 0 0 0 | 1 * * * * * {6}
xfx. ....&#xt | 2 2 2 0 | 1 0 2 0 2 1 0 0 0 | * 3 * * * * {(hhH)2}
.... xx..&#x  | 2 2 0 0 | 0 1 2 1 0 0 0 0 0 | * * 3 * * * {4}
.... .xfx&#xt | 0 2 2 2 | 0 0 0 1 2 0 2 0 1 | * * * 3 * * {(hhH)2}
..xx ....&#x  | 0 0 2 2 | 0 0 0 0 0 1 2 1 0 | * * * * 3 * {4}
...x3...x     | 0 0 0 6 | 0 0 0 0 0 0 0 3 3 | * * * * * 1 {6}

or
o...3o...     & | 12  * | 1 1  1 0 0 | 1 1 1
.o..3.o..     & |  * 12 | 0 0  1 1 1 | 0 2 1
----------------+-------+------------+------
x... ....     & |  2  0 | 6 *  * * * | 1 1 0
.... x...     & |  2  0 | * 6  * * * | 1 0 1
oo..3oo..&#x  & |  1  1 | * * 12 * * | 0 1 1
.... .x..     & |  0  2 | * *  * 6 * | 0 1 1
.oo.3.oo.&#x    |  0  2 | * *  * * 6 | 0 2 0
----------------+-------+------------+------
x...3x...     & |  6  0 | 3 3  0 0 0 | 2 * * {6}
xfx. ....&#xt & |  2  4 | 1 0  2 1 2 | * 6 * {(hhH)2}
.... xx..&#x  & |  2  2 | 0 1  2 1 0 | * * 6 {4}