Polytope of Type {2,70,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,70,4}*1120
if this polytope has a name.
Group : SmallGroup(1120,1061)
Rank : 4
Schlafli Type : {2,70,4}
Number of vertices, edges, etc : 2, 70, 140, 4
Order of s₀s₁s₂s₃ : 140
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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) :
   2-fold quotients : {2,70,2}*560
   4-fold quotients : {2,35,2}*280
   5-fold quotients : {2,14,4}*224
   7-fold quotients : {2,10,4}*160
   10-fold quotients : {2,14,2}*112
   14-fold quotients : {2,10,2}*80
   20-fold quotients : {2,7,2}*56
   28-fold quotients : {2,5,2}*40
   35-fold quotients : {2,2,4}*32
   70-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4,  9)(  5,  8)(  6,  7)( 10, 31)( 11, 37)( 12, 36)( 13, 35)( 14, 34)
( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)( 22, 26)
( 23, 25)( 39, 44)( 40, 43)( 41, 42)( 45, 66)( 46, 72)( 47, 71)( 48, 70)
( 49, 69)( 50, 68)( 51, 67)( 52, 59)( 53, 65)( 54, 64)( 55, 63)( 56, 62)
( 57, 61)( 58, 60)( 74, 79)( 75, 78)( 76, 77)( 80,101)( 81,107)( 82,106)
( 83,105)( 84,104)( 85,103)( 86,102)( 87, 94)( 88,100)( 89, 99)( 90, 98)
( 91, 97)( 92, 96)( 93, 95)(109,114)(110,113)(111,112)(115,136)(116,142)
(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)(123,135)(124,134)
(125,133)(126,132)(127,131)(128,130);;
s2 := (  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)( 17, 32)
( 18, 31)( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 25)( 26, 30)
( 27, 29)( 38, 46)( 39, 45)( 40, 51)( 41, 50)( 42, 49)( 43, 48)( 44, 47)
( 52, 67)( 53, 66)( 54, 72)( 55, 71)( 56, 70)( 57, 69)( 58, 68)( 59, 60)
( 61, 65)( 62, 64)( 73,116)( 74,115)( 75,121)( 76,120)( 77,119)( 78,118)
( 79,117)( 80,109)( 81,108)( 82,114)( 83,113)( 84,112)( 85,111)( 86,110)
( 87,137)( 88,136)( 89,142)( 90,141)( 91,140)( 92,139)( 93,138)( 94,130)
( 95,129)( 96,135)( 97,134)( 98,133)( 99,132)(100,131)(101,123)(102,122)
(103,128)(104,127)(105,126)(106,125)(107,124);;
s3 := (  3, 73)(  4, 74)(  5, 75)(  6, 76)(  7, 77)(  8, 78)(  9, 79)( 10, 80)
( 11, 81)( 12, 82)( 13, 83)( 14, 84)( 15, 85)( 16, 86)( 17, 87)( 18, 88)
( 19, 89)( 20, 90)( 21, 91)( 22, 92)( 23, 93)( 24, 94)( 25, 95)( 26, 96)
( 27, 97)( 28, 98)( 29, 99)( 30,100)( 31,101)( 32,102)( 33,103)( 34,104)
( 35,105)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)( 41,111)( 42,112)
( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)( 48,118)( 49,119)( 50,120)
( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)( 56,126)( 57,127)( 58,128)
( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,134)( 65,135)( 66,136)
( 67,137)( 68,138)( 69,139)( 70,140)( 71,141)( 72,142);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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(142)!(1,2);
s1 := Sym(142)!(  4,  9)(  5,  8)(  6,  7)( 10, 31)( 11, 37)( 12, 36)( 13, 35)
( 14, 34)( 15, 33)( 16, 32)( 17, 24)( 18, 30)( 19, 29)( 20, 28)( 21, 27)
( 22, 26)( 23, 25)( 39, 44)( 40, 43)( 41, 42)( 45, 66)( 46, 72)( 47, 71)
( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 59)( 53, 65)( 54, 64)( 55, 63)
( 56, 62)( 57, 61)( 58, 60)( 74, 79)( 75, 78)( 76, 77)( 80,101)( 81,107)
( 82,106)( 83,105)( 84,104)( 85,103)( 86,102)( 87, 94)( 88,100)( 89, 99)
( 90, 98)( 91, 97)( 92, 96)( 93, 95)(109,114)(110,113)(111,112)(115,136)
(116,142)(117,141)(118,140)(119,139)(120,138)(121,137)(122,129)(123,135)
(124,134)(125,133)(126,132)(127,131)(128,130);
s2 := Sym(142)!(  3, 11)(  4, 10)(  5, 16)(  6, 15)(  7, 14)(  8, 13)(  9, 12)
( 17, 32)( 18, 31)( 19, 37)( 20, 36)( 21, 35)( 22, 34)( 23, 33)( 24, 25)
( 26, 30)( 27, 29)( 38, 46)( 39, 45)( 40, 51)( 41, 50)( 42, 49)( 43, 48)
( 44, 47)( 52, 67)( 53, 66)( 54, 72)( 55, 71)( 56, 70)( 57, 69)( 58, 68)
( 59, 60)( 61, 65)( 62, 64)( 73,116)( 74,115)( 75,121)( 76,120)( 77,119)
( 78,118)( 79,117)( 80,109)( 81,108)( 82,114)( 83,113)( 84,112)( 85,111)
( 86,110)( 87,137)( 88,136)( 89,142)( 90,141)( 91,140)( 92,139)( 93,138)
( 94,130)( 95,129)( 96,135)( 97,134)( 98,133)( 99,132)(100,131)(101,123)
(102,122)(103,128)(104,127)(105,126)(106,125)(107,124);
s3 := Sym(142)!(  3, 73)(  4, 74)(  5, 75)(  6, 76)(  7, 77)(  8, 78)(  9, 79)
( 10, 80)( 11, 81)( 12, 82)( 13, 83)( 14, 84)( 15, 85)( 16, 86)( 17, 87)
( 18, 88)( 19, 89)( 20, 90)( 21, 91)( 22, 92)( 23, 93)( 24, 94)( 25, 95)
( 26, 96)( 27, 97)( 28, 98)( 29, 99)( 30,100)( 31,101)( 32,102)( 33,103)
( 34,104)( 35,105)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)( 41,111)
( 42,112)( 43,113)( 44,114)( 45,115)( 46,116)( 47,117)( 48,118)( 49,119)
( 50,120)( 51,121)( 52,122)( 53,123)( 54,124)( 55,125)( 56,126)( 57,127)
( 58,128)( 59,129)( 60,130)( 61,131)( 62,132)( 63,133)( 64,134)( 65,135)
( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,141)( 72,142);
poly := sub<Sym(142)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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 >;

to this polytope