It is a shallow cut Dodecahedron.
Could this be one of the "Five Pyraminxes"?
I don't remember what it's official name is, though I thought it was
the Master Pyraminx. (I will have to check my collection at home)

I only ever sow these once, in J.C Penny's, thankfully I bought one.

I always viewed this as a combining of 2 puzzles, Alexander's Star and
a round one, whose name escapes me at the moment.

My copy of this puzzle has 2 yellow and 2 red faces. I think they ran
out of colors.
This means that if I am not carefull I can appear to have 2 edges switched.
This is more apparant then real because the stickers for each face have
ridges which can be used to make the proper choice.

There are 12 faces, which can be independantly turned by 72 degrees.
Faces do not move with respect to each other.
There are 20 corners which can only be in Even Permutations.
Corners are like the cube, trios can be spun in the same direction, pairs
can be spun in opposite directions.
There are 30 edges which can only be in Even Permutations.
Edges can flipped in pairs, just like the normal cube.

group size should be
                       20     30
        30!    20!    3      2
        --- *  --- * ---  * ---
        2       2     3      2

Edge   Corn  Spin   Flip

for the supergroup, increase by a factor of
         12
        5