G-sets to counting

aliceinwonderland · #1 $Old$ December 14th, 2008, 11:43 PM

Wooden cubes of the same size are to be painted a different color on each face to make children's blocks. How many blocks can be made if eight colors of paint are available? Colors may be repeated on different faces at will. Use Burnside's formula and define G for rigid motions.

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My attempt to this problem so far

According to the application of wiki page of Burnside's lemma,

|G| = 24

One identity element which leaves all $8^{6}$ elements of X unchanged.
Six 90-degree face rotations, each of which leaves $8^{3}$ unchanged.
Three 180-degree face rotations, each of which leaves $8^{4}$ unchanged.
Eight 120-degree vertex rotations, each of which leaves $8^{2}$ unchanged. (If we give numbers for each side like 1-bottom 6-top 2-front 5-back 3-left 4-right, eight choices of same colors for 1,3,5 and eight choices of same colors for 2,4,6. )

Six 180-degree edge rotations, each of which leaves $8^{3}$ unchanged.

My question is

What is the "six 180-degree edge rotations" (hard to figure out) and what choices of colors should be given for "six 180-degree edge rotations" if we temporarily gives numbers (1,2,3,4,5,6) like above for each side?

Opalg · #2 $Old$ December 15th, 2008, 08:05 AM

The edge rotation is a 180º rotation about a line joining the midpoints of diagonally opposite edges, such as the red line in this picture.

$\setlength{\unitlength}{2mm}\begin{picture}(20,20)\put(2,2){\line(1,0){10}}\put(2,12){\line(1,0){10}}\put(7,5){\line(1,0){10}}\put(7,15){\line(1,0){10}}\put(2,2){\line(0,1){10}}\put(12,2){\line(0,1){10}}\put(7,5){\line(0,1){10}}\put(17,5){\line(0,1){10}}\put(2,2){\line(5,3){5}}\put(12,2){\line(5,3){5}}\put(2,12){\line(5,3){5}}\put(12,12){\line(5,3){5}}\put(2,7){\color{red}\line(5,1){15}}\end{picture}$

aliceinwonderland · #3 $Old$ December 15th, 2008, 01:36 PM

Thanks for your reply.

If we give numbers for each side like 1-bottom 6-top 2-front 5-back 3-left 4-right for the above figure, what would be the next position after 180-degree edge rotation?

Opalg · #4 $Old$ December 15th, 2008, 01:52 PM

Quote:

Originally Posted by aliceinwonderland $View Post$

If we give numbers for each side like 1-bottom 6-top 2-front 5-back 3-left 4-right for the above figure, what would be the next position after 180-degree edge rotation?

Try to visualise what would happen if you hold the cube at the ends of the red line and flip it upside down by twisting it through 180º.

As a permutation, the effect on the faces is (16)(23)(45). In other words, top and bottom get flipped, as do left and front, and back and right.