Polytope of Type {20,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,42}*1680b
if this polytope has a name.
Group : SmallGroup(1680,954)
Rank : 3
Schlafli Type : {20,42}
Number of vertices, edges, etc : 20, 420, 42
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 : {4,42}*336c
   7-fold quotients : {20,6}*240b
   10-fold quotients : {4,21}*168
   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,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29,115)( 30,116)
( 31,113)( 32,114)( 33,119)( 34,120)( 35,117)( 36,118)( 37,123)( 38,124)
( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)( 44,126)( 45,131)( 46,132)
( 47,129)( 48,130)( 49,135)( 50,136)( 51,133)( 52,134)( 53,139)( 54,140)
( 55,137)( 56,138)( 57, 87)( 58, 88)( 59, 85)( 60, 86)( 61, 91)( 62, 92)
( 63, 89)( 64, 90)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 99)( 70,100)
( 71, 97)( 72, 98)( 73,103)( 74,104)( 75,101)( 76,102)( 77,107)( 78,108)
( 79,105)( 80,106)( 81,111)( 82,112)( 83,109)( 84,110);;
s1 := (  1, 29)(  2, 30)(  3, 32)(  4, 31)(  5, 53)(  6, 54)(  7, 56)(  8, 55)
(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)
( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)( 24, 39)
( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 57,113)( 58,114)( 59,116)( 60,115)
( 61,137)( 62,138)( 63,140)( 64,139)( 65,133)( 66,134)( 67,136)( 68,135)
( 69,129)( 70,130)( 71,132)( 72,131)( 73,125)( 74,126)( 75,128)( 76,127)
( 77,121)( 78,122)( 79,124)( 80,123)( 81,117)( 82,118)( 83,120)( 84,119)
( 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);;
s2 := (  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, 33)( 30, 36)( 31, 35)
( 32, 34)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)( 43, 51)
( 44, 50)( 46, 48)( 57, 61)( 58, 64)( 59, 63)( 60, 62)( 65, 81)( 66, 84)
( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76)( 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)(113,117)(114,120)(115,119)(116,118)
(121,137)(122,140)(123,139)(124,138)(125,133)(126,136)(127,135)(128,134)
(130,132);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*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 ];;
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)( 29,115)
( 30,116)( 31,113)( 32,114)( 33,119)( 34,120)( 35,117)( 36,118)( 37,123)
( 38,124)( 39,121)( 40,122)( 41,127)( 42,128)( 43,125)( 44,126)( 45,131)
( 46,132)( 47,129)( 48,130)( 49,135)( 50,136)( 51,133)( 52,134)( 53,139)
( 54,140)( 55,137)( 56,138)( 57, 87)( 58, 88)( 59, 85)( 60, 86)( 61, 91)
( 62, 92)( 63, 89)( 64, 90)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 99)
( 70,100)( 71, 97)( 72, 98)( 73,103)( 74,104)( 75,101)( 76,102)( 77,107)
( 78,108)( 79,105)( 80,106)( 81,111)( 82,112)( 83,109)( 84,110);
s1 := Sym(140)!(  1, 29)(  2, 30)(  3, 32)(  4, 31)(  5, 53)(  6, 54)(  7, 56)
(  8, 55)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)
( 16, 47)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)
( 24, 39)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 57,113)( 58,114)( 59,116)
( 60,115)( 61,137)( 62,138)( 63,140)( 64,139)( 65,133)( 66,134)( 67,136)
( 68,135)( 69,129)( 70,130)( 71,132)( 72,131)( 73,125)( 74,126)( 75,128)
( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81,117)( 82,118)( 83,120)
( 84,119)( 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);
s2 := 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, 33)( 30, 36)
( 31, 35)( 32, 34)( 37, 53)( 38, 56)( 39, 55)( 40, 54)( 41, 49)( 42, 52)
( 43, 51)( 44, 50)( 46, 48)( 57, 61)( 58, 64)( 59, 63)( 60, 62)( 65, 81)
( 66, 84)( 67, 83)( 68, 82)( 69, 77)( 70, 80)( 71, 79)( 72, 78)( 74, 76)
( 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)(113,117)(114,120)(115,119)
(116,118)(121,137)(122,140)(123,139)(124,138)(125,133)(126,136)(127,135)
(128,134)(130,132);
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s1*s2*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 >;

References : None.
to this polytope