# Two Face Group

Submitted by B MacKenzie on Wed, 06/06/2007 - 09:47.

Has the Rubik cube subgroup generated by the turns of two orthogonal faces been exhaustively expanded? My computer runs out of physical memory and bogs down after 18 q turns:

Shell Classes Elements 0 1 1 1 1 4 2 3 10 3 6 24 4 15 58 5 35 140 6 85 338 7 204 816 8 493 1970 9 1189 4756 10 2863 11448 11 6862 27448 12 16324 65260 13 38550 154192 14 90192 360692 15 206898 827540 16 462893 1851345 17 992268 3968840 18 1973209 7891990 Totals 3792091 15166872

I calculate the order of the group as:

( 5! x 3^{5 }) x 7! / 2 = 73,483,200

Extrapolating the above data using an exponential factor of 2 exceeds this order by 21 q turns, so the curve must max out and begin to descend by 21 q turns. Experimentally, using a forward/backward envelope search strategy I've solved hundreds of thousands of random group elements and have not found any requiring more than 24 q turns. I suspect that 26 q turns should be sufficient to solve any element but would like to be sure.