Polytope of Type {2,45,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,45,6}*1080
if this polytope has a name.
Group : SmallGroup(1080,335)
Rank : 4
Schlafli Type : {2,45,6}
Number of vertices, edges, etc : 2, 45, 135, 6
Order of s₀s₁s₂s₃ : 90
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
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 : {2,45,2}*360, {2,15,6}*360
   5-fold quotients : {2,9,6}*216
   9-fold quotients : {2,15,2}*120
   15-fold quotients : {2,9,2}*72, {2,3,6}*72
   27-fold quotients : {2,5,2}*40
   45-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  5)(  6, 15)(  7, 17)(  8, 16)(  9, 12)( 10, 14)( 11, 13)( 18, 34)
( 19, 33)( 20, 35)( 21, 46)( 22, 45)( 23, 47)( 24, 43)( 25, 42)( 26, 44)
( 27, 40)( 28, 39)( 29, 41)( 30, 37)( 31, 36)( 32, 38)( 48, 93)( 49, 95)
( 50, 94)( 51,105)( 52,107)( 53,106)( 54,102)( 55,104)( 56,103)( 57, 99)
( 58,101)( 59,100)( 60, 96)( 61, 98)( 62, 97)( 63,124)( 64,123)( 65,125)
( 66,136)( 67,135)( 68,137)( 69,133)( 70,132)( 71,134)( 72,130)( 73,129)
( 74,131)( 75,127)( 76,126)( 77,128)( 78,109)( 79,108)( 80,110)( 81,121)
( 82,120)( 83,122)( 84,118)( 85,117)( 86,119)( 87,115)( 88,114)( 89,116)
( 90,112)( 91,111)( 92,113);;
s2 := (  3, 66)(  4, 68)(  5, 67)(  6, 63)(  7, 65)(  8, 64)(  9, 75)( 10, 77)
( 11, 76)( 12, 72)( 13, 74)( 14, 73)( 15, 69)( 16, 71)( 17, 70)( 18, 51)
( 19, 53)( 20, 52)( 21, 48)( 22, 50)( 23, 49)( 24, 60)( 25, 62)( 26, 61)
( 27, 57)( 28, 59)( 29, 58)( 30, 54)( 31, 56)( 32, 55)( 33, 82)( 34, 81)
( 35, 83)( 36, 79)( 37, 78)( 38, 80)( 39, 91)( 40, 90)( 41, 92)( 42, 88)
( 43, 87)( 44, 89)( 45, 85)( 46, 84)( 47, 86)( 93,111)( 94,113)( 95,112)
( 96,108)( 97,110)( 98,109)( 99,120)(100,122)(101,121)(102,117)(103,119)
(104,118)(105,114)(106,116)(107,115)(123,127)(124,126)(125,128)(129,136)
(130,135)(131,137)(132,133);;
s3 := ( 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)( 61,106)( 62,107)( 63,108)
( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)( 70,115)( 71,116)
( 72,117)( 73,118)( 74,119)( 75,120)( 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)( 91,136)( 92,137);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(137)!(1,2);
s1 := Sym(137)!(  4,  5)(  6, 15)(  7, 17)(  8, 16)(  9, 12)( 10, 14)( 11, 13)
( 18, 34)( 19, 33)( 20, 35)( 21, 46)( 22, 45)( 23, 47)( 24, 43)( 25, 42)
( 26, 44)( 27, 40)( 28, 39)( 29, 41)( 30, 37)( 31, 36)( 32, 38)( 48, 93)
( 49, 95)( 50, 94)( 51,105)( 52,107)( 53,106)( 54,102)( 55,104)( 56,103)
( 57, 99)( 58,101)( 59,100)( 60, 96)( 61, 98)( 62, 97)( 63,124)( 64,123)
( 65,125)( 66,136)( 67,135)( 68,137)( 69,133)( 70,132)( 71,134)( 72,130)
( 73,129)( 74,131)( 75,127)( 76,126)( 77,128)( 78,109)( 79,108)( 80,110)
( 81,121)( 82,120)( 83,122)( 84,118)( 85,117)( 86,119)( 87,115)( 88,114)
( 89,116)( 90,112)( 91,111)( 92,113);
s2 := Sym(137)!(  3, 66)(  4, 68)(  5, 67)(  6, 63)(  7, 65)(  8, 64)(  9, 75)
( 10, 77)( 11, 76)( 12, 72)( 13, 74)( 14, 73)( 15, 69)( 16, 71)( 17, 70)
( 18, 51)( 19, 53)( 20, 52)( 21, 48)( 22, 50)( 23, 49)( 24, 60)( 25, 62)
( 26, 61)( 27, 57)( 28, 59)( 29, 58)( 30, 54)( 31, 56)( 32, 55)( 33, 82)
( 34, 81)( 35, 83)( 36, 79)( 37, 78)( 38, 80)( 39, 91)( 40, 90)( 41, 92)
( 42, 88)( 43, 87)( 44, 89)( 45, 85)( 46, 84)( 47, 86)( 93,111)( 94,113)
( 95,112)( 96,108)( 97,110)( 98,109)( 99,120)(100,122)(101,121)(102,117)
(103,119)(104,118)(105,114)(106,116)(107,115)(123,127)(124,126)(125,128)
(129,136)(130,135)(131,137)(132,133);
s3 := Sym(137)!( 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)( 61,106)( 62,107)
( 63,108)( 64,109)( 65,110)( 66,111)( 67,112)( 68,113)( 69,114)( 70,115)
( 71,116)( 72,117)( 73,118)( 74,119)( 75,120)( 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)( 91,136)( 92,137);
poly := sub<Sym(137)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope