Polytope of Type {4,10,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,10,14}*1120
Also Known As : {{4,10|2},{10,14|2}}. if this polytope has another name.
Group : SmallGroup(1120,998)
Rank : 4
Schlafli Type : {4,10,14}
Number of vertices, edges, etc : 4, 20, 70, 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 : {2,10,14}*560
   5-fold quotients : {4,2,14}*224
   7-fold quotients : {4,10,2}*160
   10-fold quotients : {4,2,7}*112, {2,2,14}*112
   14-fold quotients : {2,10,2}*80
   20-fold quotients : {2,2,7}*56
   28-fold quotients : {2,5,2}*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, 72)(  3, 73)(  4, 74)(  5, 75)(  6, 76)(  7, 77)(  8, 99)
(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)( 16, 93)
( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22, 85)( 23, 86)( 24, 87)
( 25, 88)( 26, 89)( 27, 90)( 28, 91)( 29, 78)( 30, 79)( 31, 80)( 32, 81)
( 33, 82)( 34, 83)( 35, 84)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,111)( 42,112)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)( 48,139)
( 49,140)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)( 56,133)
( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126)( 64,113)
( 65,114)( 66,115)( 67,116)( 68,117)( 69,118)( 70,119);;
s2 := (  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);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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)!( 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, 72)(  3, 73)(  4, 74)(  5, 75)(  6, 76)(  7, 77)
(  8, 99)(  9,100)( 10,101)( 11,102)( 12,103)( 13,104)( 14,105)( 15, 92)
( 16, 93)( 17, 94)( 18, 95)( 19, 96)( 20, 97)( 21, 98)( 22, 85)( 23, 86)
( 24, 87)( 25, 88)( 26, 89)( 27, 90)( 28, 91)( 29, 78)( 30, 79)( 31, 80)
( 32, 81)( 33, 82)( 34, 83)( 35, 84)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,111)( 42,112)( 43,134)( 44,135)( 45,136)( 46,137)( 47,138)
( 48,139)( 49,140)( 50,127)( 51,128)( 52,129)( 53,130)( 54,131)( 55,132)
( 56,133)( 57,120)( 58,121)( 59,122)( 60,123)( 61,124)( 62,125)( 63,126)
( 64,113)( 65,114)( 66,115)( 67,116)( 68,117)( 69,118)( 70,119);
s2 := 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);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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.
to this polytope