Polytope of Type {4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*1280b
if this polytope has a name.
Group : SmallGroup(1280,1116442)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 32, 320, 160
Order of s₀s₁s₂ : 10
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10}*640b
   4-fold quotients : {4,5}*320, {4,10}*320a, {4,10}*320b
   8-fold quotients : {4,5}*160
   32-fold quotients : {2,10}*40
   64-fold quotients : {2,5}*20
   160-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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