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Polytope of Type {10,4,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,4,14}*1120
Also Known As : {{10,4|2},{4,14|2}}. if this polytope has another name.
Group : SmallGroup(1120,998)
Rank : 4
Schlafli Type : {10,4,14}
Number of vertices, edges, etc : 10, 20, 28, 14
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 : {10,2,14}*560
   4-fold quotients : {5,2,14}*280, {10,2,7}*280
   5-fold quotients : {2,4,14}*224
   7-fold quotients : {10,4,2}*160
   8-fold quotients : {5,2,7}*140
   10-fold quotients : {2,2,14}*112
   14-fold quotients : {10,2,2}*80
   20-fold quotients : {2,2,7}*56
   28-fold quotients : {5,2,2}*40
   35-fold quotients : {2,4,2}*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,  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)( 71,113)( 72,114)( 73,115)( 74,116)
( 75,117)( 76,118)( 77,119)( 78,106)( 79,107)( 80,108)( 81,109)( 82,110)
( 83,111)( 84,112)( 85,134)( 86,135)( 87,136)( 88,137)( 89,138)( 90,139)
( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)( 96,131)( 97,132)( 98,133)
( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126);;
s2 := (  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);;
s3 := (  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, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)( 92, 93)
( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)(109,111)
(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)(129,133)
(130,132)(134,135)(136,140)(137,139);;
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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,  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)( 71,113)( 72,114)( 73,115)
( 74,116)( 75,117)( 76,118)( 77,119)( 78,106)( 79,107)( 80,108)( 81,109)
( 82,110)( 83,111)( 84,112)( 85,134)( 86,135)( 87,136)( 88,137)( 89,138)
( 90,139)( 91,140)( 92,127)( 93,128)( 94,129)( 95,130)( 96,131)( 97,132)
( 98,133)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126);
s2 := 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);
s3 := 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, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85, 86)( 87, 91)( 88, 90)
( 92, 93)( 94, 98)( 95, 97)( 99,100)(101,105)(102,104)(106,107)(108,112)
(109,111)(113,114)(115,119)(116,118)(120,121)(122,126)(123,125)(127,128)
(129,133)(130,132)(134,135)(136,140)(137,139);
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*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
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