Polytope of Type {140,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {140,6}*1680b
if this polytope has a name.
Group : SmallGroup(1680,955)
Rank : 3
Schlafli Type : {140,6}
Number of vertices, edges, etc : 140, 420, 6
Order of s₀s₁s₂ : 105
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
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) :
   5-fold quotients : {28,6}*336b
   7-fold quotients : {20,6}*240b
   35-fold quotients : {4,6}*48b
   70-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)( 10, 24)
( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29,115)( 30,116)
( 31,113)( 32,114)( 33,139)( 34,140)( 35,137)( 36,138)( 37,135)( 38,136)
( 39,133)( 40,134)( 41,131)( 42,132)( 43,129)( 44,130)( 45,127)( 46,128)
( 47,125)( 48,126)( 49,123)( 50,124)( 51,121)( 52,122)( 53,119)( 54,120)
( 55,117)( 56,118)( 57, 87)( 58, 88)( 59, 85)( 60, 86)( 61,111)( 62,112)
( 63,109)( 64,110)( 65,107)( 66,108)( 67,105)( 68,106)( 69,103)( 70,104)
( 71,101)( 72,102)( 73, 99)( 74,100)( 75, 97)( 76, 98)( 77, 95)( 78, 96)
( 79, 93)( 80, 94)( 81, 91)( 82, 92)( 83, 89)( 84, 90);;
s1 := (  1, 33)(  2, 34)(  3, 36)(  4, 35)(  5, 29)(  6, 30)(  7, 32)(  8, 31)
(  9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 49)( 14, 50)( 15, 52)( 16, 51)
( 17, 45)( 18, 46)( 19, 48)( 20, 47)( 21, 41)( 22, 42)( 23, 44)( 24, 43)
( 25, 37)( 26, 38)( 27, 40)( 28, 39)( 57,117)( 58,118)( 59,120)( 60,119)
( 61,113)( 62,114)( 63,116)( 64,115)( 65,137)( 66,138)( 67,140)( 68,139)
( 69,133)( 70,134)( 71,136)( 72,135)( 73,129)( 74,130)( 75,132)( 76,131)
( 77,125)( 78,126)( 79,128)( 80,127)( 81,121)( 82,122)( 83,124)( 84,123)
( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 93,109)( 94,110)( 95,112)( 96,111)
( 97,105)( 98,106)( 99,108)(100,107)(103,104);;
s2 := (  2,  4)(  6,  8)( 10, 12)( 14, 16)( 18, 20)( 22, 24)( 26, 28)( 30, 32)
( 34, 36)( 38, 40)( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 58, 60)( 62, 64)
( 66, 68)( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 86, 88)( 90, 92)( 94, 96)
( 98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)(126,128)
(130,132)(134,136)(138,140);;
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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(140)!(  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)
( 10, 24)( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29,115)
( 30,116)( 31,113)( 32,114)( 33,139)( 34,140)( 35,137)( 36,138)( 37,135)
( 38,136)( 39,133)( 40,134)( 41,131)( 42,132)( 43,129)( 44,130)( 45,127)
( 46,128)( 47,125)( 48,126)( 49,123)( 50,124)( 51,121)( 52,122)( 53,119)
( 54,120)( 55,117)( 56,118)( 57, 87)( 58, 88)( 59, 85)( 60, 86)( 61,111)
( 62,112)( 63,109)( 64,110)( 65,107)( 66,108)( 67,105)( 68,106)( 69,103)
( 70,104)( 71,101)( 72,102)( 73, 99)( 74,100)( 75, 97)( 76, 98)( 77, 95)
( 78, 96)( 79, 93)( 80, 94)( 81, 91)( 82, 92)( 83, 89)( 84, 90);
s1 := Sym(140)!(  1, 33)(  2, 34)(  3, 36)(  4, 35)(  5, 29)(  6, 30)(  7, 32)
(  8, 31)(  9, 53)( 10, 54)( 11, 56)( 12, 55)( 13, 49)( 14, 50)( 15, 52)
( 16, 51)( 17, 45)( 18, 46)( 19, 48)( 20, 47)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 37)( 26, 38)( 27, 40)( 28, 39)( 57,117)( 58,118)( 59,120)
( 60,119)( 61,113)( 62,114)( 63,116)( 64,115)( 65,137)( 66,138)( 67,140)
( 68,139)( 69,133)( 70,134)( 71,136)( 72,135)( 73,129)( 74,130)( 75,132)
( 76,131)( 77,125)( 78,126)( 79,128)( 80,127)( 81,121)( 82,122)( 83,124)
( 84,123)( 85, 89)( 86, 90)( 87, 92)( 88, 91)( 93,109)( 94,110)( 95,112)
( 96,111)( 97,105)( 98,106)( 99,108)(100,107)(103,104);
s2 := Sym(140)!(  2,  4)(  6,  8)( 10, 12)( 14, 16)( 18, 20)( 22, 24)( 26, 28)
( 30, 32)( 34, 36)( 38, 40)( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 58, 60)
( 62, 64)( 66, 68)( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 86, 88)( 90, 92)
( 94, 96)( 98,100)(102,104)(106,108)(110,112)(114,116)(118,120)(122,124)
(126,128)(130,132)(134,136)(138,140);
poly := sub<Sym(140)|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*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope