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Polytope of Type {10,14,4}

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