Polytope of Type {9,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1785)
Rank : 3
Schlafli Type : {9,6}
Number of vertices, edges, etc : 108, 324, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
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 : {9,6}*432, {3,6}*432
   4-fold quotients : {9,6}*324a
   9-fold quotients : {3,6}*144
   12-fold quotients : {9,6}*108, {3,6}*108
   27-fold quotients : {3,6}*48
   36-fold quotients : {9,2}*36, {3,6}*36
   54-fold quotients : {3,3}*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=<s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
      36 facets:
         36 of {9}*18
      54 vertex figures:
         54 of {6}*12
   P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      42 vertex figures:
         33 of {6}*12
         9 of {2}*4
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 3.
      24 facets:
         24 of {9}*18
      36 vertex figures:
         36 of {6}*12
   P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 4.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      18 facets:
         18 of {9}*18
      27 vertex figures:
         27 of {6}*12
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1> of order 6.
      12 facets:
         12 of {9}*18
      18 vertex figures:
         18 of {6}*12

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