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Polytope of Type {4,14,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,14,10}*1120
Also Known As : {{4,14|2},{14,10|2}}. if this polytope has another name.
Group : SmallGroup(1120,998)
Rank : 4
Schlafli Type : {4,14,10}
Number of vertices, edges, etc : 4, 28, 70, 10
Order of s0s1s2s3 : 140
Order of s0s1s2s3s2s1 : 2
Special Properties :
   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 : {2,14,10}*560
   5-fold quotients : {4,14,2}*224
   7-fold quotients : {4,2,10}*160
   10-fold quotients : {2,14,2}*112
   14-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {2,7,2}*56
   28-fold quotients : {2,2,5}*40
   35-fold quotients : {4,2,2}*32
   70-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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, 78)
(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 85)( 16, 91)
( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 92)( 23, 98)( 24, 97)
( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 99)( 30,105)( 31,104)( 32,103)
( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)( 40,109)
( 41,108)( 42,107)( 43,113)( 44,119)( 45,118)( 46,117)( 47,116)( 48,115)
( 49,114)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)( 56,121)
( 57,127)( 58,133)( 59,132)( 60,131)( 61,130)( 62,129)( 63,128)( 64,134)
( 65,140)( 66,139)( 67,138)( 68,137)( 69,136)( 70,135);;
s2 := (  1,  2)(  3,  7)(  4,  6)(  8, 30)(  9, 29)( 10, 35)( 11, 34)( 12, 33)
( 13, 32)( 14, 31)( 15, 23)( 16, 22)( 17, 28)( 18, 27)( 19, 26)( 20, 25)
( 21, 24)( 36, 37)( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)( 46, 69)
( 47, 68)( 48, 67)( 49, 66)( 50, 58)( 51, 57)( 52, 63)( 53, 62)( 54, 61)
( 55, 60)( 56, 59)( 71, 72)( 73, 77)( 74, 76)( 78,100)( 79, 99)( 80,105)
( 81,104)( 82,103)( 83,102)( 84,101)( 85, 93)( 86, 92)( 87, 98)( 88, 97)
( 89, 96)( 90, 95)( 91, 94)(106,107)(108,112)(109,111)(113,135)(114,134)
(115,140)(116,139)(117,138)(118,137)(119,136)(120,128)(121,127)(122,133)
(123,132)(124,131)(125,130)(126,129);;
s3 := (  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)( 15, 29)
( 16, 30)( 17, 31)( 18, 32)( 19, 33)( 20, 34)( 21, 35)( 36, 43)( 37, 44)
( 38, 45)( 39, 46)( 40, 47)( 41, 48)( 42, 49)( 50, 64)( 51, 65)( 52, 66)
( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 71, 78)( 72, 79)( 73, 80)( 74, 81)
( 75, 82)( 76, 83)( 77, 84)( 85, 99)( 86,100)( 87,101)( 88,102)( 89,103)
( 90,104)( 91,105)(106,113)(107,114)(108,115)(109,116)(110,117)(111,118)
(112,119)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)(126,140);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 ];;
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, 78)(  9, 84)( 10, 83)( 11, 82)( 12, 81)( 13, 80)( 14, 79)( 15, 85)
( 16, 91)( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 92)( 23, 98)
( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28, 93)( 29, 99)( 30,105)( 31,104)
( 32,103)( 33,102)( 34,101)( 35,100)( 36,106)( 37,112)( 38,111)( 39,110)
( 40,109)( 41,108)( 42,107)( 43,113)( 44,119)( 45,118)( 46,117)( 47,116)
( 48,115)( 49,114)( 50,120)( 51,126)( 52,125)( 53,124)( 54,123)( 55,122)
( 56,121)( 57,127)( 58,133)( 59,132)( 60,131)( 61,130)( 62,129)( 63,128)
( 64,134)( 65,140)( 66,139)( 67,138)( 68,137)( 69,136)( 70,135);
s2 := Sym(140)!(  1,  2)(  3,  7)(  4,  6)(  8, 30)(  9, 29)( 10, 35)( 11, 34)
( 12, 33)( 13, 32)( 14, 31)( 15, 23)( 16, 22)( 17, 28)( 18, 27)( 19, 26)
( 20, 25)( 21, 24)( 36, 37)( 38, 42)( 39, 41)( 43, 65)( 44, 64)( 45, 70)
( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 58)( 51, 57)( 52, 63)( 53, 62)
( 54, 61)( 55, 60)( 56, 59)( 71, 72)( 73, 77)( 74, 76)( 78,100)( 79, 99)
( 80,105)( 81,104)( 82,103)( 83,102)( 84,101)( 85, 93)( 86, 92)( 87, 98)
( 88, 97)( 89, 96)( 90, 95)( 91, 94)(106,107)(108,112)(109,111)(113,135)
(114,134)(115,140)(116,139)(117,138)(118,137)(119,136)(120,128)(121,127)
(122,133)(123,132)(124,131)(125,130)(126,129);
s3 := Sym(140)!(  1,  8)(  2,  9)(  3, 10)(  4, 11)(  5, 12)(  6, 13)(  7, 14)
( 15, 29)( 16, 30)( 17, 31)( 18, 32)( 19, 33)( 20, 34)( 21, 35)( 36, 43)
( 37, 44)( 38, 45)( 39, 46)( 40, 47)( 41, 48)( 42, 49)( 50, 64)( 51, 65)
( 52, 66)( 53, 67)( 54, 68)( 55, 69)( 56, 70)( 71, 78)( 72, 79)( 73, 80)
( 74, 81)( 75, 82)( 76, 83)( 77, 84)( 85, 99)( 86,100)( 87,101)( 88,102)
( 89,103)( 90,104)( 91,105)(106,113)(107,114)(108,115)(109,116)(110,117)
(111,118)(112,119)(120,134)(121,135)(122,136)(123,137)(124,138)(125,139)
(126,140);
poly := sub<Sym(140)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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 >; 
 
References : None.
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