Polytope of Type {42,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,20}*1680b
if this polytope has a name.
Group : SmallGroup(1680,954)
Rank : 3
Schlafli Type : {42,20}
Number of vertices, edges, etc : 42, 420, 20
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 :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {42,4}*336c
   7-fold quotients : {6,20}*240b
   10-fold quotients : {21,4}*168
   35-fold quotients : {6,4}*48b
   70-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope