Polytope of Type {4,4,5}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,5}*1280
Also Known As : {{4,4|2},{4,5|4}}. if this polytope has another name.
Group : SmallGroup(1280,1116442)
Rank : 4
Schlafli Type : {4,4,5}
Number of vertices, edges, etc : 4, 64, 80, 40
Order of s₀s₁s₂s₃ : 20
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 : {2,4,5}*640
   4-fold quotients : {2,4,5}*320
   16-fold quotients : {4,2,5}*80
   32-fold quotients : {2,2,5}*40
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 13)( 10, 14)( 11, 16)( 12, 15)
( 17, 21)( 18, 22)( 19, 24)( 20, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)
( 33, 37)( 34, 38)( 35, 40)( 36, 39)( 41, 45)( 42, 46)( 43, 48)( 44, 47)
( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 57, 61)( 58, 62)( 59, 64)( 60, 63)
( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 73, 77)( 74, 78)( 75, 80)( 76, 79)
( 81, 85)( 82, 86)( 83, 88)( 84, 87)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 97,101)( 98,102)( 99,104)(100,103)(105,109)(106,110)(107,112)(108,111)
(113,117)(114,118)(115,120)(116,119)(121,125)(122,126)(123,128)(124,127);;
s1 := (  1, 59)(  2, 60)(  3, 57)(  4, 58)(  5, 63)(  6, 64)(  7, 61)(  8, 62)
(  9, 51)( 10, 52)( 11, 49)( 12, 50)( 13, 55)( 14, 56)( 15, 53)( 16, 54)
( 17, 43)( 18, 44)( 19, 41)( 20, 42)( 21, 47)( 22, 48)( 23, 45)( 24, 46)
( 25, 35)( 26, 36)( 27, 33)( 28, 34)( 29, 39)( 30, 40)( 31, 37)( 32, 38)
( 65,123)( 66,124)( 67,121)( 68,122)( 69,127)( 70,128)( 71,125)( 72,126)
( 73,115)( 74,116)( 75,113)( 76,114)( 77,119)( 78,120)( 79,117)( 80,118)
( 81,107)( 82,108)( 83,105)( 84,106)( 85,111)( 86,112)( 87,109)( 88,110)
( 89, 99)( 90,100)( 91, 97)( 92, 98)( 93,103)( 94,104)( 95,101)( 96,102);;
s2 := (  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)( 16,104)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)( 24, 64)
( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)( 32, 96)
( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)( 40, 80)
( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)( 56,120)
( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)( 88,128);;
s3 := ( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 71,127)( 72,128)
( 73,113)( 74,114)( 75,115)( 76,116)( 77,117)( 78,118)( 79,119)( 80,120)
( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)( 88,104)
( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)( 96,112);;
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*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(128)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 13)( 10, 14)( 11, 16)
( 12, 15)( 17, 21)( 18, 22)( 19, 24)( 20, 23)( 25, 29)( 26, 30)( 27, 32)
( 28, 31)( 33, 37)( 34, 38)( 35, 40)( 36, 39)( 41, 45)( 42, 46)( 43, 48)
( 44, 47)( 49, 53)( 50, 54)( 51, 56)( 52, 55)( 57, 61)( 58, 62)( 59, 64)
( 60, 63)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 73, 77)( 74, 78)( 75, 80)
( 76, 79)( 81, 85)( 82, 86)( 83, 88)( 84, 87)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 97,101)( 98,102)( 99,104)(100,103)(105,109)(106,110)(107,112)
(108,111)(113,117)(114,118)(115,120)(116,119)(121,125)(122,126)(123,128)
(124,127);
s1 := Sym(128)!(  1, 59)(  2, 60)(  3, 57)(  4, 58)(  5, 63)(  6, 64)(  7, 61)
(  8, 62)(  9, 51)( 10, 52)( 11, 49)( 12, 50)( 13, 55)( 14, 56)( 15, 53)
( 16, 54)( 17, 43)( 18, 44)( 19, 41)( 20, 42)( 21, 47)( 22, 48)( 23, 45)
( 24, 46)( 25, 35)( 26, 36)( 27, 33)( 28, 34)( 29, 39)( 30, 40)( 31, 37)
( 32, 38)( 65,123)( 66,124)( 67,121)( 68,122)( 69,127)( 70,128)( 71,125)
( 72,126)( 73,115)( 74,116)( 75,113)( 76,114)( 77,119)( 78,120)( 79,117)
( 80,118)( 81,107)( 82,108)( 83,105)( 84,106)( 85,111)( 86,112)( 87,109)
( 88,110)( 89, 99)( 90,100)( 91, 97)( 92, 98)( 93,103)( 94,104)( 95,101)
( 96,102);
s2 := Sym(128)!(  9, 97)( 10, 98)( 11, 99)( 12,100)( 13,101)( 14,102)( 15,103)
( 16,104)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 61)( 22, 62)( 23, 63)
( 24, 64)( 25, 89)( 26, 90)( 27, 91)( 28, 92)( 29, 93)( 30, 94)( 31, 95)
( 32, 96)( 33, 73)( 34, 74)( 35, 75)( 36, 76)( 37, 77)( 38, 78)( 39, 79)
( 40, 80)( 49,113)( 50,114)( 51,115)( 52,116)( 53,117)( 54,118)( 55,119)
( 56,120)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)
( 88,128);
s3 := Sym(128)!( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 65,121)( 66,122)( 67,123)( 68,124)( 69,125)( 70,126)( 71,127)
( 72,128)( 73,113)( 74,114)( 75,115)( 76,116)( 77,117)( 78,118)( 79,119)
( 80,120)( 81, 97)( 82, 98)( 83, 99)( 84,100)( 85,101)( 86,102)( 87,103)
( 88,104)( 89,105)( 90,106)( 91,107)( 92,108)( 93,109)( 94,110)( 95,111)
( 96,112);
poly := sub<Sym(128)|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*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;

References : None.
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