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Polytope of Type {70,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {70,4}*560
Also Known As : {70,4|2}. if this polytope has another name.
Group : SmallGroup(560,160)
Rank : 3
Schlafli Type : {70,4}
Number of vertices, edges, etc : 70, 140, 4
Order of s0s1s2 : 140
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {70,4,2} of size 1120
Vertex Figure Of :
   {2,70,4} of size 1120
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {70,2}*280
   4-fold quotients : {35,2}*140
   5-fold quotients : {14,4}*112
   7-fold quotients : {10,4}*80
   10-fold quotients : {14,2}*56
   14-fold quotients : {10,2}*40
   20-fold quotients : {7,2}*28
   28-fold quotients : {5,2}*20
   35-fold quotients : {2,4}*16
   70-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {140,4}*1120, {70,8}*1120
   3-fold covers : {70,12}*1680, {210,4}*1680a
Permutation Representation (GAP) :
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)( 12, 32)
( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)( 20, 24)
( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)( 46, 68)
( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)
( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)( 80,104)
( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)( 88, 96)
( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)(114,140)
(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)(122,132)
(123,131)(124,130)(125,129)(126,128);;
s1 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)( 57, 58)
( 59, 63)( 60, 62)( 71,114)( 72,113)( 73,119)( 74,118)( 75,117)( 76,116)
( 77,115)( 78,107)( 79,106)( 80,112)( 81,111)( 82,110)( 83,109)( 84,108)
( 85,135)( 86,134)( 87,140)( 88,139)( 89,138)( 90,137)( 91,136)( 92,128)
( 93,127)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,121)(100,120)
(101,126)(102,125)(103,124)(104,123)(105,122);;
s2 := (  1, 71)(  2, 72)(  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);;
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, 
s1*s2*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*s1*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*s1*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*s1*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*s1*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(140)!(  2,  7)(  3,  6)(  4,  5)(  8, 29)(  9, 35)( 10, 34)( 11, 33)
( 12, 32)( 13, 31)( 14, 30)( 15, 22)( 16, 28)( 17, 27)( 18, 26)( 19, 25)
( 20, 24)( 21, 23)( 37, 42)( 38, 41)( 39, 40)( 43, 64)( 44, 70)( 45, 69)
( 46, 68)( 47, 67)( 48, 66)( 49, 65)( 50, 57)( 51, 63)( 52, 62)( 53, 61)
( 54, 60)( 55, 59)( 56, 58)( 72, 77)( 73, 76)( 74, 75)( 78, 99)( 79,105)
( 80,104)( 81,103)( 82,102)( 83,101)( 84,100)( 85, 92)( 86, 98)( 87, 97)
( 88, 96)( 89, 95)( 90, 94)( 91, 93)(107,112)(108,111)(109,110)(113,134)
(114,140)(115,139)(116,138)(117,137)(118,136)(119,135)(120,127)(121,133)
(122,132)(123,131)(124,130)(125,129)(126,128);
s1 := Sym(140)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)
( 42, 45)( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 57, 58)( 59, 63)( 60, 62)( 71,114)( 72,113)( 73,119)( 74,118)( 75,117)
( 76,116)( 77,115)( 78,107)( 79,106)( 80,112)( 81,111)( 82,110)( 83,109)
( 84,108)( 85,135)( 86,134)( 87,140)( 88,139)( 89,138)( 90,137)( 91,136)
( 92,128)( 93,127)( 94,133)( 95,132)( 96,131)( 97,130)( 98,129)( 99,121)
(100,120)(101,126)(102,125)(103,124)(104,123)(105,122);
s2 := Sym(140)!(  1, 71)(  2, 72)(  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);
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, 
s1*s2*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*s1*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*s1*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*s1*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*s1*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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