Polytope of Type {12,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*864f
if this polytope has a name.
Group : SmallGroup(864,2282)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,12,2} of size 1728
Vertex Figure Of :
   {2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*432f
   3-fold quotients : {12,4}*288
   4-fold quotients : {6,12}*216b
   6-fold quotients : {6,4}*144
   12-fold quotients : {6,4}*72
   27-fold quotients : {4,4}*32
   54-fold quotients : {2,4}*16, {4,2}*16
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,12}*1728f, {12,24}*1728g, {24,12}*1728i, {24,12}*1728k, {12,24}*1728m
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2> of order 2.
      18 facets:
         18 of {12}*24
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s2*s1*s0*s1*s0*s2*s1*s0*s1> of order 3.
      12 facets:
         12 of {12}*24
      12 vertex figures:
         12 of {12}*24

Permutation Representation (GAP) :

s0 := (  1, 55)(  2, 56)(  3, 57)(  4, 63)(  5, 61)(  6, 62)(  7, 59)(  8, 60)(  9, 58)( 10, 73)( 11, 74)( 12, 75)( 13, 81)( 14, 79)( 15, 80)( 16, 77)( 17, 78)( 18, 76)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)( 24, 71)( 25, 68)( 26, 69)( 27, 67)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,100)( 38,101)( 39,102)( 40,108)( 41,106)( 42,107)( 43,104)( 44,105)( 45,103)( 46, 91)( 47, 92)( 48, 93)( 49, 99)( 50, 97)( 51, 98)( 52, 95)( 53, 96)( 54, 94);;
s1 := (  1, 13)(  2, 15)(  3, 14)(  4,  6)(  7, 23)(  8, 22)(  9, 24)( 11, 12)( 16, 19)( 17, 21)( 18, 20)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 33)( 34, 50)( 35, 49)( 36, 51)( 38, 39)( 43, 46)( 44, 48)( 45, 47)( 52, 54)( 55, 94)( 56, 96)( 57, 95)( 58, 87)( 59, 86)( 60, 85)( 61,104)( 62,103)( 63,105)( 64, 91)( 65, 93)( 66, 92)( 67, 82)( 68, 84)( 69, 83)( 70,100)( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 89)( 77, 88)( 78, 90)( 79,108)( 80,107)( 81,106);;
s2 := (  2,  3)(  4,  5)(  7,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 32)( 34, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 50)( 41, 49)( 42, 51)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 59)( 61, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 77)( 68, 76)( 69, 78)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)( 96,105)( 97,108)( 98,107)( 99,106);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(108)!(  1, 55)(  2, 56)(  3, 57)(  4, 63)(  5, 61)(  6, 62)(  7, 59)(  8, 60)(  9, 58)( 10, 73)( 11, 74)( 12, 75)( 13, 81)( 14, 79)( 15, 80)( 16, 77)( 17, 78)( 18, 76)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)( 24, 71)( 25, 68)( 26, 69)( 27, 67)( 28, 82)( 29, 83)( 30, 84)( 31, 90)( 32, 88)( 33, 89)( 34, 86)( 35, 87)( 36, 85)( 37,100)( 38,101)( 39,102)( 40,108)( 41,106)( 42,107)( 43,104)( 44,105)( 45,103)( 46, 91)( 47, 92)( 48, 93)( 49, 99)( 50, 97)( 51, 98)( 52, 95)( 53, 96)( 54, 94);
s1 := Sym(108)!(  1, 13)(  2, 15)(  3, 14)(  4,  6)(  7, 23)(  8, 22)(  9, 24)( 11, 12)( 16, 19)( 17, 21)( 18, 20)( 25, 27)( 28, 40)( 29, 42)( 30, 41)( 31, 33)( 34, 50)( 35, 49)( 36, 51)( 38, 39)( 43, 46)( 44, 48)( 45, 47)( 52, 54)( 55, 94)( 56, 96)( 57, 95)( 58, 87)( 59, 86)( 60, 85)( 61,104)( 62,103)( 63,105)( 64, 91)( 65, 93)( 66, 92)( 67, 82)( 68, 84)( 69, 83)( 70,100)( 71,102)( 72,101)( 73, 97)( 74, 99)( 75, 98)( 76, 89)( 77, 88)( 78, 90)( 79,108)( 80,107)( 81,106);
s2 := Sym(108)!(  2,  3)(  4,  5)(  7,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 23)( 14, 22)( 15, 24)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 31, 32)( 34, 36)( 37, 46)( 38, 48)( 39, 47)( 40, 50)( 41, 49)( 42, 51)( 43, 54)( 44, 53)( 45, 52)( 56, 57)( 58, 59)( 61, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 77)( 68, 76)( 69, 78)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 85, 86)( 88, 90)( 91,100)( 92,102)( 93,101)( 94,104)( 95,103)( 96,105)( 97,108)( 98,107)( 99,106);
poly := sub<Sym(108)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1 >;

References : None.
to this polytope

Polytope of Type {12,12}

Twisty Puzzle