Polytope of Type {6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20}*1080
if this polytope has a name.
Group : SmallGroup(1080,283)
Rank : 3
Schlafli Type : {6,20}
Number of vertices, edges, etc : 27, 270, 90
Order of s₀s₁s₂ : 60
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {6,20}*360
   5-fold quotients : {6,4}*216
   15-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 46, 91)
( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)( 54, 99)
( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)( 62,122)
( 63,123)( 64,124)( 65,125)( 66,126)( 67,127)( 68,128)( 69,129)( 70,130)
( 71,131)( 72,132)( 73,133)( 74,134)( 75,135)( 76,106)( 77,107)( 78,108)
( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)( 85,115)( 86,116)
( 87,117)( 88,118)( 89,119)( 90,120);;
s1 := (  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)(  8, 57)
(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)( 16, 62)
( 17, 61)( 18, 63)( 19, 74)( 20, 73)( 21, 75)( 22, 71)( 23, 70)( 24, 72)
( 25, 68)( 26, 67)( 27, 69)( 28, 65)( 29, 64)( 30, 66)( 31, 78)( 32, 77)
( 33, 76)( 34, 90)( 35, 89)( 36, 88)( 37, 87)( 38, 86)( 39, 85)( 40, 84)
( 41, 83)( 42, 82)( 43, 81)( 44, 80)( 45, 79)( 92, 93)( 94,103)( 95,105)
( 96,104)( 97,100)( 98,102)( 99,101)(106,107)(109,119)(110,118)(111,120)
(112,116)(113,115)(114,117)(121,123)(124,135)(125,134)(126,133)(127,132)
(128,131)(129,130);;
s2 := (  1,  5)(  2,  4)(  3,  6)(  7, 14)(  8, 13)(  9, 15)( 10, 11)( 16, 50)
( 17, 49)( 18, 51)( 19, 47)( 20, 46)( 21, 48)( 22, 59)( 23, 58)( 24, 60)
( 25, 56)( 26, 55)( 27, 57)( 28, 53)( 29, 52)( 30, 54)( 31, 95)( 32, 94)
( 33, 96)( 34, 92)( 35, 91)( 36, 93)( 37,104)( 38,103)( 39,105)( 40,101)
( 41,100)( 42,102)( 43, 98)( 44, 97)( 45, 99)( 61, 66)( 62, 65)( 63, 64)
( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 76,109)( 77,111)( 78,110)( 79,106)
( 80,108)( 81,107)( 82,118)( 83,120)( 84,119)( 85,115)( 86,117)( 87,116)
( 88,112)( 89,114)( 90,113)(121,126)(122,125)(123,124)(127,135)(128,134)
(129,133)(130,132);;
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*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(135)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 46, 91)( 47, 92)( 48, 93)( 49, 94)( 50, 95)( 51, 96)( 52, 97)( 53, 98)
( 54, 99)( 55,100)( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)( 61,121)
( 62,122)( 63,123)( 64,124)( 65,125)( 66,126)( 67,127)( 68,128)( 69,129)
( 70,130)( 71,131)( 72,132)( 73,133)( 74,134)( 75,135)( 76,106)( 77,107)
( 78,108)( 79,109)( 80,110)( 81,111)( 82,112)( 83,113)( 84,114)( 85,115)
( 86,116)( 87,117)( 88,118)( 89,119)( 90,120);
s1 := Sym(135)!(  1, 46)(  2, 48)(  3, 47)(  4, 58)(  5, 60)(  6, 59)(  7, 55)
(  8, 57)(  9, 56)( 10, 52)( 11, 54)( 12, 53)( 13, 49)( 14, 51)( 15, 50)
( 16, 62)( 17, 61)( 18, 63)( 19, 74)( 20, 73)( 21, 75)( 22, 71)( 23, 70)
( 24, 72)( 25, 68)( 26, 67)( 27, 69)( 28, 65)( 29, 64)( 30, 66)( 31, 78)
( 32, 77)( 33, 76)( 34, 90)( 35, 89)( 36, 88)( 37, 87)( 38, 86)( 39, 85)
( 40, 84)( 41, 83)( 42, 82)( 43, 81)( 44, 80)( 45, 79)( 92, 93)( 94,103)
( 95,105)( 96,104)( 97,100)( 98,102)( 99,101)(106,107)(109,119)(110,118)
(111,120)(112,116)(113,115)(114,117)(121,123)(124,135)(125,134)(126,133)
(127,132)(128,131)(129,130);
s2 := Sym(135)!(  1,  5)(  2,  4)(  3,  6)(  7, 14)(  8, 13)(  9, 15)( 10, 11)
( 16, 50)( 17, 49)( 18, 51)( 19, 47)( 20, 46)( 21, 48)( 22, 59)( 23, 58)
( 24, 60)( 25, 56)( 26, 55)( 27, 57)( 28, 53)( 29, 52)( 30, 54)( 31, 95)
( 32, 94)( 33, 96)( 34, 92)( 35, 91)( 36, 93)( 37,104)( 38,103)( 39,105)
( 40,101)( 41,100)( 42,102)( 43, 98)( 44, 97)( 45, 99)( 61, 66)( 62, 65)
( 63, 64)( 67, 75)( 68, 74)( 69, 73)( 70, 72)( 76,109)( 77,111)( 78,110)
( 79,106)( 80,108)( 81,107)( 82,118)( 83,120)( 84,119)( 85,115)( 86,117)
( 87,116)( 88,112)( 89,114)( 90,113)(121,126)(122,125)(123,124)(127,135)
(128,134)(129,133)(130,132);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1 >;

References : None.
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