Polytope of Type {6,105}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,105}*1680
if this polytope has a name.
Group : SmallGroup(1680,955)
Rank : 3
Schlafli Type : {6,105}
Number of vertices, edges, etc : 8, 420, 140
Order of s₀s₁s₂ : 140
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   5-fold quotients : {6,21}*336
   7-fold quotients : {6,15}*240
   12-fold quotients : {2,35}*140
   35-fold quotients : {6,3}*48
   60-fold quotients : {2,7}*28
   70-fold quotients : {3,3}*24
   84-fold quotients : {2,5}*20
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope