Polytope of Type {6,9}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*1296b
if this polytope has a name.
Group : SmallGroup(1296,1785)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 72, 324, 108
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,9}*432, {6,3}*432
   4-fold quotients : {6,9}*324a
   9-fold quotients : {6,3}*144
   12-fold quotients : {6,9}*108, {6,3}*108
   27-fold quotients : {6,3}*48
   36-fold quotients : {2,9}*36, {6,3}*36
   54-fold quotients : {3,3}*24
   108-fold quotients : {2,3}*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*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2> of order 2.
      54 facets:
         54 of {6}*12
      36 vertex figures:
         36 of {9}*18
   P/N, where N=<s0*s1*s0*s1> of order 3.
      42 facets:
         9 of {2}*4
         33 of {6}*12
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 3.
      36 facets:
         36 of {6}*12
      24 vertex figures:
         24 of {9}*18
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      27 facets:
         27 of {6}*12
      18 vertex figures:
         18 of {9}*18
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2> of order 4.
      27 facets:
         27 of {6}*12
      18 vertex figures:
         18 of {9}*18
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s1*s2> of order 6.
      18 facets:
         18 of {6}*12
      12 vertex figures:
         12 of {9}*18

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