Polytope of Type {14,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,20}*560
Also Known As : {14,20|2}. if this polytope has another name.
Group : SmallGroup(560,120)
Rank : 3
Schlafli Type : {14,20}
Number of vertices, edges, etc : 14, 140, 20
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 :
   {14,20,2} of size 1120
Vertex Figure Of :
   {2,14,20} of size 1120
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,10}*280
   5-fold quotients : {14,4}*112
   7-fold quotients : {2,20}*80
   10-fold quotients : {14,2}*56
   14-fold quotients : {2,10}*40
   20-fold quotients : {7,2}*28
   28-fold quotients : {2,5}*20
   35-fold quotients : {2,4}*16
   70-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,40}*1120, {28,20}*1120
   3-fold covers : {14,60}*1680, {42,20}*1680a
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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