Polytope of Type {12,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296f
if this polytope has a name.
Group : SmallGroup(1296,1790)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s0s1s2 : 9
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {12,6}*432d
   4-fold quotients : {6,6}*324a
   9-fold quotients : {12,6}*144d
   12-fold quotients : {6,6}*108
   27-fold quotients : {4,6}*48b
   54-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      36 facets:
         18 of {6}*12
         18 of {12}*24
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1> of order 3.
      30 facets:
         18 of {4}*8
         12 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 3.
      18 facets:
         18 of {12}*24
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s1> of order 6.
      24 facets:
         18 of {2}*4
         6 of {12}*24
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      18 facets:
         6 of {6}*12
         3 of {12}*24
         9 of {4}*8
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      12 facets:
         6 of {6}*12
         6 of {12}*24
      18 vertex figures:
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1> of order 9.
      10 facets:
         4 of {12}*24
         6 of {4}*8
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s1*s0*s1*s2> of order 9.
      6 facets:
         6 of {12}*24
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1, s0*s2*s1*s0*s1*s0*s1*s0*s1*s2> of order 9.
      14 facets:
         12 of {4}*8
         2 of {12}*24
      12 vertex figures:
         12 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s0*s1*s0*s1*s2, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 18.
      8 facets:
         2 of {6}*12
         6 of {4}*8
      6 vertex figures:
         6 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)( 42, 84)( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,107)( 54,108)( 55,105)( 56,106)( 57,103)( 58,104)( 59,101)( 60,102)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)( 32, 63)( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 93)( 86, 94)( 87, 96)( 88, 95)( 91, 92)( 97,101)( 98,102)( 99,104)(100,103)(107,108);;
s2 := (  2,  4)(  6,  8)( 10, 12)( 13, 25)( 14, 28)( 15, 27)( 16, 26)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 21, 33)( 22, 36)( 23, 35)( 24, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)( 82,108)( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(108)!(  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 27)( 14, 28)( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)( 42, 84)( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 99)( 50,100)( 51, 97)( 52, 98)( 53,107)( 54,108)( 55,105)( 56,106)( 57,103)( 58,104)( 59,101)( 60,102)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90);
s1 := Sym(108)!(  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 57)( 14, 58)( 15, 60)( 16, 59)( 17, 53)( 18, 54)( 19, 56)( 20, 55)( 21, 49)( 22, 50)( 23, 52)( 24, 51)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 61)( 30, 62)( 31, 64)( 32, 63)( 33, 69)( 34, 70)( 35, 72)( 36, 71)( 75, 76)( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 85, 93)( 86, 94)( 87, 96)( 88, 95)( 91, 92)( 97,101)( 98,102)( 99,104)(100,103)(107,108);
s2 := Sym(108)!(  2,  4)(  6,  8)( 10, 12)( 13, 25)( 14, 28)( 15, 27)( 16, 26)( 17, 29)( 18, 32)( 19, 31)( 20, 30)( 21, 33)( 22, 36)( 23, 35)( 24, 34)( 37, 49)( 38, 52)( 39, 51)( 40, 50)( 41, 53)( 42, 56)( 43, 55)( 44, 54)( 45, 57)( 46, 60)( 47, 59)( 48, 58)( 62, 64)( 66, 68)( 70, 72)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,101)( 78,104)( 79,103)( 80,102)( 81,105)( 82,108)( 83,107)( 84,106)( 86, 88)( 90, 92)( 94, 96);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*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 >;
References : None.
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