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Polytope of Type {36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {36,6}*1296i
if this polytope has a name.
Group : SmallGroup(1296,1783)
Rank : 3
Schlafli Type : {36,6}
Number of vertices, edges, etc : 108, 324, 18
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 : {18,6}*324b
   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.
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, 83)( 38, 84)( 39, 81)( 40, 82)( 41, 79)( 42, 80)
( 43, 77)( 44, 78)( 45, 75)( 46, 76)( 47, 73)( 48, 74)( 49,107)( 50,108)
( 51,105)( 52,106)( 53,103)( 54,104)( 55,101)( 56,102)( 57, 99)( 58,100)
( 59, 97)( 60, 98)( 61, 95)( 62, 96)( 63, 93)( 64, 94)( 65, 91)( 66, 92)
( 67, 89)( 68, 90)( 69, 87)( 70, 88)( 71, 85)( 72, 86);;
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)( 73, 81)( 74, 82)( 75, 84)( 76, 83)
( 79, 80)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 99,100)(101,105)
(102,106)(103,108)(104,107);;
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*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*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, 83)( 38, 84)( 39, 81)( 40, 82)( 41, 79)
( 42, 80)( 43, 77)( 44, 78)( 45, 75)( 46, 76)( 47, 73)( 48, 74)( 49,107)
( 50,108)( 51,105)( 52,106)( 53,103)( 54,104)( 55,101)( 56,102)( 57, 99)
( 58,100)( 59, 97)( 60, 98)( 61, 95)( 62, 96)( 63, 93)( 64, 94)( 65, 91)
( 66, 92)( 67, 89)( 68, 90)( 69, 87)( 70, 88)( 71, 85)( 72, 86);
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)( 73, 81)( 74, 82)( 75, 84)
( 76, 83)( 79, 80)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 95, 96)( 99,100)
(101,105)(102,106)(103,108)(104,107);
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*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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