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Subgroup enumeration

Submitted by B MacKenzie on Sun, 04/12/2009 - 15:07.

I've been playing around with the Rubik's cube subgroup generated by the turns: (U U' D D' L2 R2 F2 B2) which I refer to as the D4h cube subgroup after the symmetry invariance of the generator set. This, I believe, is the subgroup employed by Kociemba in his two phase algorithm. Anyway, I have performed a partial enumeration of the subgroup and its coset space. I was wondering if anyone might be able to confirm these numbers as a check on my methodology. 

Six Face/D4h Coset Enumeration (q turns)
Depth  Cosets     Total
 0         1         1
 1         4         5
 2        34        39
 3       312       351
 4      2772      3123
 5     24996     28119
 6    225949    254068
 7   2017078   2271146

D4h cube group enumeration, 8 gen (U U' D D' L2 R2 F2 B2)
Depth Class(D4h) Elements   Total
 0         1         1         1  
 1         2         8         9  
 2         7        48        57  
 3        25       284       341  
 4       124      1678      2019 
 5       648      9664     11683  
 6      3523     54475     66158  
 7     19006    299960    366118 
 8    100741   1602352   1968470  
 9    518843   8279732  10248202 
10   2571647  41089158  51337360  

D4h cube group enumeration, 10 gen (U U' D D' L2 R2 F2 B2 U2 D2)
Depth Class(D4h) Elements   Total
 0         1         1         1  
 1         3        10        11  
 2        12        73        84  
 3        48       514       598  
 4       262      3515      4113 
 5      1550     22984     27097  
 6      9245    142982    170079  
 7     54578    859946   1030025  
 8    309714   4922028   5952053 
 9   1681312  26815882  32767935
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Numbers

Submitted by rokicki on Mon, 04/13/2009 - 14:09.
Okay, let's start with the easiest one first, which is the last (cube group enumeration, using all 10 generators).

These numbers already appear in several publications and web sites, and appear to disagree already at level 2, where the total should be 78, not 84. See for example

http://www.risc.uni-linz.ac.at/publications/download/risc_3122/uppernew3.ps

http://cubezzz.homelinux.org/drupal/?q=node/view/51

It is possible I am somehow misunderstanding what the table is supposed to show, and if that is the case I apologize.

Your first table appears correct as far as you've done it; see the first link.

Anyway, glad you are working in this group; I think there are many opportunities yet unexplored.
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Thank you for the links. I s

Submitted by B MacKenzie on Mon, 04/13/2009 - 16:37.

Thank you for the links. I suspected the numbers were available somewhere on the web but couldn't find them.

I found the flaw in my algorithm. I'm use to working in the q turn realm where adding a turn to a state at depth n can only produce duplicates of states at depth n-1 and depth n+1. In the half turn realm duplicates also occur at depth n. So, I was counting some states twice. After eliminating this error my numbers now match those of Reid. This error does not effect the 8 generator enumeration, however, so I believe the numbers in the 8 generator case are OK.

Thanks again.

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