NB. Very inefficent wrt memory and somewhat inefficent wrt speed,
NB. but no need to optimize on a modern machine.

NB. a cube is a list of 6 sides (each atoms): 
NB. left front right back top bottom.
NB. the first four of these sides are visible on the sides in a tower.
NB. a towers is a list of 4 cubes.

NB. colorings.
input =: >;: 'pgpygr rprrgy ppryyg rrygpy' 

NB. all 24 rotations of a cube.
gens =: (i.6),'NSKTFL'i.'FSLTKN',:'SKTNFL'
rots =: ([:~.[:/:~[:,/{"1/)^:_~ gens
   
NB. all 24 rotations of each 4 cube.
posn =: rots {"_ 1 input

NB. stack a list of 4 lists of 24 cubes to a tower of 4 cubes in each 24^4 way,
NB. order of 4 cubes doesn't matter.
stack =: [:>[:>[:,[:{[:<"1<"1
towers =: stack posn

NB. check if a cube is good or bad -- rank 2, gives bool atom.
good =: [: (*./"1) 4 ({."1) 4 = [: (#@:~."1) |:"2
solns =: (good # ]) towers

NB. each solution gives 8 trivially equivalent solutions because you can 
NB. rotate all cubes simultanously around the vertical axis or twice around
NB. the horizontal axis.  (this may be less than 8 for degenerate cubes.)
NB. rota y gives all 8 rotations of a tower y
rott =: (2{gens)&({"_ 1)
rotu =: ({~1{gens)&({"_ 1)
rota =: ([: ,/ [: rott^:(<4) rotu^:(<2))"2
NB. verify that each rotation of each solution is indeed among the solutions.
assert =: 3 :'assert. y'
assert *./ solns e.~ ,/ rota solns
NB. now normalize the solutions by changing a tower to the 
NB. lexicographically first out of these eight variants,
NB. and weed out repetitions.
norml =: ([:{.[:/:~rota)"2
solns1 =: ~./:~ norml solns

NB. output the solutions. there's just one if you use the sample
NB. cube colorings in the post.
echo solns1
echo #solns1

NB. end

##</code><code>##

ggrypp
ypgrrr
ryppyg
prygyr
1

##</code><code>##

   <"2 solns
+------+------+------+------+------+------+------+------+
|ggyrpp|grygpp|yggrpp|yrggpp|ggrypp|gyrgpp|rggypp|ryggpp|
|pyrgrr|pgryrr|rypgrr|rgpyrr|ypgrrr|yrgprr|gpyrrr|gryprr|
|yrppgy|yppryg|prypyg|ppyrgy|ryppyg|rppygy|pyrpgy|ppryyg|
|rpgyry|rygpyr|gpryyr|gyrpry|prygyr|pgyrry|yrpgry|ygpryr|
+------+------+------+------+------+------+------+------+

##</code><code>##

   ,/"2,"1&' ' |:"2 (1 3 0 2){"1 solns
gyrp rgpy gpyr yrpg 
rgpy gyrp gpyr yrpg 
gyrp rgpy yrpg gpyr 
rgpy gyrp yrpg gpyr 
gpyr yrpg gyrp rgpy 
yrpg gpyr gyrp rgpy 
gpyr yrpg rgpy gyrp 
yrpg gpyr rgpy gyrp