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Order of group of n X n X n Rubik cube, under assumptions not-s, m, not-i.
7

%I #18 Jul 04 2022 08:00:38

%S 1,3674160,43252003274489856000,

%T 707195371192426622240452051915172831683411968000000000,

%U 2582636272886959379162819698174683585918088940054237132144778804568925405184000000000000000

%N Order of group of n X n X n Rubik cube, under assumptions not-s, m, not-i.

%C The three possible assumptions considered here are the following:

%C s (for n odd) indicates that we are working in the "supergroup" and so take account of twists of the face centers.

%C m (for n > 3) indicates that the pieces are marked so that we take account of the permutation of the identically-colored pieces on a face.

%C i (for n > 3) indicates that we are working in the theoretical invisible group and solve the pieces on the interior of the cube as well as the exterior. It is assumed that the M and S traits apply to the interior pieces as if they were on the exterior of a smaller cube.

%D Dan Hoey, posting to Cube Lovers List, Jun 24, 1987.

%D Rowley, Chris, The group of the Hungarian magic cube, in Algebraic structures and applications (Nedlands, 1980), pp. 33-43, Lecture Notes in Pure and Appl. Math., 74, Dekker, New York, 1982.

%H Alan Bawden, <a href="ftp://ftp.ai.mit.edu/pub/cube-lovers/cube-mail-6.gz">Cube Lovers Archive, Part 6</a>

%F a(1)=1; a(2)=7!*3^6; a(3)=8!*3^7*12!*2^10; a(n)=a(n-2)*24!*(24!/2)^(n-3). - _Herbert Kociemba_, Dec 08 2016

%p f := proc(n) local A,B,C,D,E,F,G; if n mod 2 = 1 then A := (n-1)/2; B := 1; C := 1; D := 0; E := (n+1)*(n-3)/4; F := 0; G := 0; else A := n/2; B := 1; C := 0; D := 0; E := n*(n-2)/4; F := 1; G := 0; fi; (2^A*((8!/2)*3^7)^B*((12!/2)*2^11)^C*((4^6)/2)^D*(24!/2)^E)/(24^F*((24^6)/2)^G); end;

%t f[1]=1;f[2]=7!3^6;f[3]=8!3^7 12!2^10;f[n_]:=f[n-2]*24!(24!/2)^(n-3); Array[f,5] (* _Herbert Kociemba_, Dec 08 2016 *)

%t f[1]=1;f[n_]:=7!3^6(6*24!!)^(s=Mod[n,2])24!^(r=(n-s)/2-1)(24!/2)^(r(r+s)); Array[f,5] (* _Herbert Kociemba_, Jul 03 2022 *)

%Y See A007458, A054434, A075152, A074914, A080658, A080659, A080660, A080661, A080662 for other versions.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Mar 01 2003