"8 Dice arranged as a Cube" Face-Sum Problem I found this here:

Sum Problem
Given eight dice. Build a $2\times 2\times2$ cube, so that the sum of the points on each side is the same.
$\hskip2.7in$
Here is one of  20 736 solutions with the sum 14.
  You find more at the German magazine "Bild der Wissenschaft 3-1980". 

Now I have three (Question 1 moved here) questions:


*

* Is $14$ the only possible face sum? At least, in the example given, it seems to related to the fact, that on every face two dice-pairs show up, having $n$ and $7-n$ pips. Is this necessary? Sufficient it is... 

*How do they get $20736$? This is the dimension of the related group and factors to $2^8\times 3^4$, the number of group elements, right? 

i. I can get $2^3$, by the following: In the example given, you can split along the $xy$ ($yz,zx$) plane and then interchange the $2$ blocks of $4$ dice. Wlog, mirroring at $xy$ commutes with $yz$ (both just invert the $z$ resp. $x$ coordinate, right), so we get $2^3$ group lements. $$ $$
  ii. The factor $3$ looks related to rotations throught the diagonals. But without my role playing set at hand, I can't work that out. $$ $$
  iii. Would rolling the overall die around an axis also count, since back and front always shows a "rotated" pattern? This would give six $90^\circ$-rotations and three $180^\circ$-rotations, $9=3^2$ in total. 
  $$ \\ $$
  Where do the missing $2^5\times 3^2$ come from?


*Is the reference given, online available?
EDIT
And to not make
tehshrike sad
again,
here's the special question for $D4$:
What face sum is possible, so that the sum of the points on each side
is the same, when you pile up 4 D4's to a pyramid (plus the octahedron mentioned by Henning) and how many representations, would such a pyramid
have?
Thanks

Regarding your reference request:
The site of the magazine offers many of their articles online starting from 1997, so you cannot obtain the 1980 edition online (although you can likely buy a used print version).
Most good libraries in German-speaking countries do have this magazine, so, depending on your country, you could go directly go to the library, try to get an inter-library loan or contact friends in German-speaking countries to scan the appropriate pages for you. 
Of course, the article will be in German.
