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Polytope of Type {20,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,6}*1080
if this polytope has a name.
Group : SmallGroup(1080,283)
Rank : 3
Schlafli Type : {20,6}
Number of vertices, edges, etc : 90, 270, 27
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
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 : {20,6}*360
   5-fold quotients : {4,6}*216
   15-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)( 17, 18)
( 19, 28)( 20, 30)( 21, 29)( 22, 25)( 23, 27)( 24, 26)( 32, 33)( 34, 43)
( 35, 45)( 36, 44)( 37, 40)( 38, 42)( 39, 41)( 46, 91)( 47, 93)( 48, 92)
( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)( 55, 97)( 56, 99)
( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,106)( 62,108)( 63,107)( 64,118)
( 65,120)( 66,119)( 67,115)( 68,117)( 69,116)( 70,112)( 71,114)( 72,113)
( 73,109)( 74,111)( 75,110)( 76,121)( 77,123)( 78,122)( 79,133)( 80,135)
( 81,134)( 82,130)( 83,132)( 84,131)( 85,127)( 86,129)( 87,128)( 88,124)
( 89,126)( 90,125);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)( 16, 49)
( 17, 51)( 18, 50)( 19, 46)( 20, 48)( 21, 47)( 22, 58)( 23, 60)( 24, 59)
( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 94)( 32, 96)
( 33, 95)( 34, 91)( 35, 93)( 36, 92)( 37,103)( 38,105)( 39,104)( 40,100)
( 41,102)( 42,101)( 43, 97)( 44, 99)( 45, 98)( 61, 65)( 62, 64)( 63, 66)
( 67, 74)( 68, 73)( 69, 75)( 70, 71)( 76,111)( 77,110)( 78,109)( 79,108)
( 80,107)( 81,106)( 82,120)( 83,119)( 84,118)( 85,117)( 86,116)( 87,115)
( 88,114)( 89,113)( 90,112)(121,125)(122,124)(123,126)(127,134)(128,133)
(129,135)(130,131);;
s2 := (  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)(  8, 23)
(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)( 46,106)
( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)( 54,114)
( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)( 62, 92)
( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)( 70,100)
( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)( 78,123)
( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)( 86,131)
( 87,132)( 88,133)( 89,134)( 90,135);;
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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(135)!(  2,  3)(  4, 13)(  5, 15)(  6, 14)(  7, 10)(  8, 12)(  9, 11)
( 17, 18)( 19, 28)( 20, 30)( 21, 29)( 22, 25)( 23, 27)( 24, 26)( 32, 33)
( 34, 43)( 35, 45)( 36, 44)( 37, 40)( 38, 42)( 39, 41)( 46, 91)( 47, 93)
( 48, 92)( 49,103)( 50,105)( 51,104)( 52,100)( 53,102)( 54,101)( 55, 97)
( 56, 99)( 57, 98)( 58, 94)( 59, 96)( 60, 95)( 61,106)( 62,108)( 63,107)
( 64,118)( 65,120)( 66,119)( 67,115)( 68,117)( 69,116)( 70,112)( 71,114)
( 72,113)( 73,109)( 74,111)( 75,110)( 76,121)( 77,123)( 78,122)( 79,133)
( 80,135)( 81,134)( 82,130)( 83,132)( 84,131)( 85,127)( 86,129)( 87,128)
( 88,124)( 89,126)( 90,125);
s1 := Sym(135)!(  1,  4)(  2,  6)(  3,  5)(  7, 13)(  8, 15)(  9, 14)( 11, 12)
( 16, 49)( 17, 51)( 18, 50)( 19, 46)( 20, 48)( 21, 47)( 22, 58)( 23, 60)
( 24, 59)( 25, 55)( 26, 57)( 27, 56)( 28, 52)( 29, 54)( 30, 53)( 31, 94)
( 32, 96)( 33, 95)( 34, 91)( 35, 93)( 36, 92)( 37,103)( 38,105)( 39,104)
( 40,100)( 41,102)( 42,101)( 43, 97)( 44, 99)( 45, 98)( 61, 65)( 62, 64)
( 63, 66)( 67, 74)( 68, 73)( 69, 75)( 70, 71)( 76,111)( 77,110)( 78,109)
( 79,108)( 80,107)( 81,106)( 82,120)( 83,119)( 84,118)( 85,117)( 86,116)
( 87,115)( 88,114)( 89,113)( 90,112)(121,125)(122,124)(123,126)(127,134)
(128,133)(129,135)(130,131);
s2 := Sym(135)!(  1, 16)(  2, 17)(  3, 18)(  4, 19)(  5, 20)(  6, 21)(  7, 22)
(  8, 23)(  9, 24)( 10, 25)( 11, 26)( 12, 27)( 13, 28)( 14, 29)( 15, 30)
( 46,106)( 47,107)( 48,108)( 49,109)( 50,110)( 51,111)( 52,112)( 53,113)
( 54,114)( 55,115)( 56,116)( 57,117)( 58,118)( 59,119)( 60,120)( 61, 91)
( 62, 92)( 63, 93)( 64, 94)( 65, 95)( 66, 96)( 67, 97)( 68, 98)( 69, 99)
( 70,100)( 71,101)( 72,102)( 73,103)( 74,104)( 75,105)( 76,121)( 77,122)
( 78,123)( 79,124)( 80,125)( 81,126)( 82,127)( 83,128)( 84,129)( 85,130)
( 86,131)( 87,132)( 88,133)( 89,134)( 90,135);
poly := sub<Sym(135)|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, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0 >; 
 
References : None.
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