Polytope of Type {10,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,28}*560
Also Known As : {10,28|2}. if this polytope has another name.
Group : SmallGroup(560,121)
Rank : 3
Schlafli Type : {10,28}
Number of vertices, edges, etc : 10, 140, 28
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 :
   {10,28,2} of size 1120
Vertex Figure Of :
   {2,10,28} of size 1120
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,14}*280
   5-fold quotients : {2,28}*112
   7-fold quotients : {10,4}*80
   10-fold quotients : {2,14}*56
   14-fold quotients : {10,2}*40
   20-fold quotients : {2,7}*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 : {10,56}*1120, {20,28}*1120
   3-fold covers : {30,28}*1680a, {10,84}*1680
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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