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I can add a bit of additional information. The 16 positions

8q from Start and 9q from B can be reduced to 4 positions

unique up to Q2-conjugacy. As I have discussed before,

it is still difficult to claim that the 4 positions are really

"different" without further analysis because of the possibility

that the positions are variations within a commuting subsequence

of moves.

I don't really have a Q2-conjugacy program. It would be easy to

make one, but I don't have time so I used my M-conjugacy program.

Recall that Q2={i,b,bb,bbb,rrv,rrbv,ttv,ttbv}, where b, r, and

t are whole cube rotations of the Back, Right, and Top faces,

respectively, and v is the central inversion. For 12 of the

16 positions X the program reports Symm(X)={i}, which is to say

m'Xm is not equal X for any m in M except the identity. Obviously,

the same is true for all m in Q2 since Q2 is a subgroup of M. We

have |Q2|=8, so |{m'Xm | m in Q2}=6. Therefore, the 12 positions

for which Symm(X)={i} form two Q2-conjugacy classes.

Using the M-conjugacy program for the other 4 positions is trickier,

but only slightly so. For the other 4 positions, the M-conjugacy

program reports Symm(X)=HX, where HX={i,bb,rr,tt,v,bbv,rrv,ttv}.

But HX is not a subgroup of Q2, and what we need is sort of

"Symm(X) with respect to Q2", which I will call Symm(X/Q2).

(A better notation is probably available). It is easy to see

that Symm(X/Q2)=(Symm(X) intersect Q2), and we have

(HX intersect Q2)={i,ttv,bb,rrv}. This subgroup is called

HQ2 in Dan's taxonomy.

We have |Q2|=8 and |HQ2|=4, so |{m'Xm | m in HQ2}|=2 when Symm(X/Q2)=HQ2. Therefore, the last 4 positions form two Q2-conjugacy classes. = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = Robert G. Bryan (Jerry Bryan) (304) 293-5192 Associate Director, WVNET (304) 293-5540 fax 837 Chestnut Ridge Road BRYAN@WVNVM Morgantown, WV 26505 BRYAN@WVNVM.WVNET.EDU