I am having difficulty understanding the third stage of Thistlethwaite's algorithm.
I am using http://www.jaapsch.net/puzzles/thistle.htm.
The third stage has a factor of [8!/(4!4!)]2 *2*3. This corresponds to the fact that the edges in the L and R faces are placed in their correct slices, the corners are put into their correct tetrads, the parity of the edge permutation (and hence the corners too) is made even, and the total twist of each tetrad is fixed.
If I understand correctly, a tetrad is formed from the path a corner travels using only moves in
<L2,R2,F2,B2,U2,D2>. This forms two tetrads, each consisting of four corners, and each corner can then be placed into its respective tetrad.
Then the edge slices can be arranged and the parity determined. This counts up to
C(8, 4)2 × 2 = 9,800
The size of stage three, however, is 29,400 which means I am off by a factor of 3.
What I don't understand is the meaning of tetrad twist and how it generates a factor of 3.