Questions?
See the FAQ
or other info.

Polytope of Type {4,70}

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