Polytope of Type {6,36}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1296j
if this polytope has a name.
Group : SmallGroup(1296,1784)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 18, 324, 108
Order of s0s1s2 : 3
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 : {6,12}*432d
   4-fold quotients : {6,18}*324c
   9-fold quotients : {6,12}*144d
   12-fold quotients : {6,6}*108
   27-fold quotients : {6,4}*48b
   54-fold quotients : {3,4}*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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      54 facets:
         54 of {6}*12
      12 vertex figures:
         6 of {36}*72
         6 of {18}*36
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2> of order 3.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72

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