Polytope of Type {4,105}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,105}*840
if this polytope has a name.
Group : SmallGroup(840,155)
Rank : 3
Schlafli Type : {4,105}
Number of vertices, edges, etc : 4, 210, 105
Order of s₀s₁s₂ : 105
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,105,2} of size 1680
Vertex Figure Of :
   {2,4,105} of size 1680
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {4,21}*168
   7-fold quotients : {4,15}*120
   35-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,105}*1680, {4,210}*1680b, {4,210}*1680c
Permutation Representation (GAP) :

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

References : None.
to this polytope