I've been fooling around with symmetric Rubik cube patterns: "Cross" patterns, involving corners only; "Check" patterns, involving edges only; and "Dots" patterns, involving edges and corners.

By rotating the eight corners a quarter turn about one of the cube axes a cube state with a cross on four faces is produced. A second such state may be produced by rotating about a different cube axes:

R U B R U R D R U' D B' U' B' D' B' R' D' B'
R B D R B R F R B' F D' B' D' F' D' R' F' D'

By recursively forming binary products from these two generators a group of 24 cube states is produced composed of all the different pure cross patterns plus the identity element. This Cross group is composed of five conjugate classes:

Normally, we think of Rubik's cube symmetry in terms of a particular position x.  The symmetry of x can be defined as Symm(x), where Symm(x) is the set all symmetries m in M such that x^m=x, and where M is the group of 48 symmetries of the cube.  By the closure property, Symm(x) is a subgroup of M, and there are 98 possible such subgroups.  I wish to expand this definition of symmetry to encompass an entire subgroup of the Cube group G.

The primary motivation for the expanded definition is to provide a standard mechanism to deal with the symmetry of subgroups of the Cube group G such as H=<U,R>.  I'm probably going to make things more complicated than they need to be.  However, I don't recall seeing a general discussion of group symmetries before, neither on Cube-Lovers nor on this forum.

I have written an optimal solver console program for the quarter turn metric in C which compiles under Linux and Windows. The target subgroup for the pruning table is the permutations of <U,D,R2,F2,L2,B2> which have an even corner parity, lets call it H'. I experimented with another type of pruning table which does not store the distance to H' for each coset element but the moves which reduce the distance to H'. Because there are 12 possible moves in QTM, a 16bit word is enough for each coset element - a move reduces the distance to H' by one or increases it by one. The table fits in about 583 MB of RAM.

At last, I have completed my God's algorithm calculation for the 4x4x4 supercube squares group, in the single-slice half-turns only metric. It was found to have 47,968,768 antipodes, many more elements than are in the entire 3x3x3 supercube squares group. (OK, it's only 11,992,192 antipodes after reducing the group by four cube orientations.) The analysis showed that the elements in the group can always be solved using no more than 20 half-turns, only one more half-turn than the number required if the four centers for each face are considered indistinguishable. (The 4x4x4 supercube differs from the usual 4x4x4 cube in that all 24 centers are considered to be distinguishable from one another.)