Polytope of Type {9,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,12}*1296d
if this polytope has a name.
Group : SmallGroup(1296,1790)
Rank : 3
Schlafli Type : {9,12}
Number of vertices, edges, etc : 54, 324, 72
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {3,12}*432
   4-fold quotients : {9,6}*324b
   9-fold quotients : {3,12}*144
   12-fold quotients : {3,6}*108
   27-fold quotients : {3,4}*48
   36-fold quotients : {3,6}*36
   54-fold quotients : {3,4}*24
   108-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 2.
      36 facets:
         36 of {9}*18
      36 vertex figures:
         18 of {12}*24
         18 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      36 facets:
         36 of {9}*18
      27 vertex figures:
         27 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1> of order 3.
      24 facets:
         24 of {9}*18
      30 vertex figures:
         12 of {12}*24
         18 of {4}*8
   P/N, where N=<s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      18 vertex figures:
         18 of {12}*24
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {9}*18
      24 vertex figures:
         6 of {12}*24
         18 of {2}*4
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         9 of {4}*8
         3 of {12}*24
         6 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1, s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      9 vertex figures:
         9 of {12}*24
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      15 vertex figures:
         6 of {12}*24
         9 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      12 facets:
         12 of {9}*18
      12 vertex figures:
         6 of {12}*24
         6 of {6}*12

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