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Polytope of Type {4,8,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,8,20}*1280a
if this polytope has a name.
Group : SmallGroup(1280,201129)
Rank : 4
Schlafli Type : {4,8,20}
Number of vertices, edges, etc : 4, 16, 80, 20
Order of s0s1s2s3 : 40
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 : {4,4,20}*640
   4-fold quotients : {2,4,20}*320, {4,2,20}*320, {4,4,10}*320
   5-fold quotients : {4,8,4}*256a
   8-fold quotients : {2,2,20}*160, {2,4,10}*160, {4,2,10}*160
   10-fold quotients : {4,4,4}*128
   16-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {2,2,5}*40
   40-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
   80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)(  8, 48)
(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)( 24, 69)
( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)( 32, 77)
( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)
( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)( 88,128)
( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)( 96,136)
( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)(104,149)
(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)(112,157)
(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)(120,155);;
s1 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 21, 26)( 22, 27)( 23, 28)
( 24, 29)( 25, 30)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 61, 66)
( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81,101)( 82,102)( 83,103)( 84,104)
( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)( 92,117)
( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)
(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)
(137,152)(138,153)(139,154)(140,155);;
s2 := (  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)(  8, 89)
(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)( 16, 96)
( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,106)( 22,110)( 23,109)( 24,108)
( 25,107)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,116)( 32,120)
( 33,119)( 34,118)( 35,117)( 36,111)( 37,115)( 38,114)( 39,113)( 40,112)
( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)( 48,129)
( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)( 56,136)
( 57,140)( 58,139)( 59,138)( 60,137)( 61,146)( 62,150)( 63,149)( 64,148)
( 65,147)( 66,141)( 67,145)( 68,144)( 69,143)( 70,142)( 71,156)( 72,160)
( 73,159)( 74,158)( 75,157)( 76,151)( 77,155)( 78,154)( 79,153)( 80,152);;
s3 := (  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)(  8, 46)
(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)( 16, 58)
( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 68)( 22, 67)( 23, 66)( 24, 70)
( 25, 69)( 26, 63)( 27, 62)( 28, 61)( 29, 65)( 30, 64)( 31, 78)( 32, 77)
( 33, 76)( 34, 80)( 35, 79)( 36, 73)( 37, 72)( 38, 71)( 39, 75)( 40, 74)
( 81,133)( 82,132)( 83,131)( 84,135)( 85,134)( 86,138)( 87,137)( 88,136)
( 89,140)( 90,139)( 91,123)( 92,122)( 93,121)( 94,125)( 95,124)( 96,128)
( 97,127)( 98,126)( 99,130)(100,129)(101,158)(102,157)(103,156)(104,160)
(105,159)(106,153)(107,152)(108,151)(109,155)(110,154)(111,148)(112,147)
(113,146)(114,150)(115,149)(116,143)(117,142)(118,141)(119,145)(120,144);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(160)!(  1, 41)(  2, 42)(  3, 43)(  4, 44)(  5, 45)(  6, 46)(  7, 47)
(  8, 48)(  9, 49)( 10, 50)( 11, 51)( 12, 52)( 13, 53)( 14, 54)( 15, 55)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 66)( 22, 67)( 23, 68)
( 24, 69)( 25, 70)( 26, 61)( 27, 62)( 28, 63)( 29, 64)( 30, 65)( 31, 76)
( 32, 77)( 33, 78)( 34, 79)( 35, 80)( 36, 71)( 37, 72)( 38, 73)( 39, 74)
( 40, 75)( 81,121)( 82,122)( 83,123)( 84,124)( 85,125)( 86,126)( 87,127)
( 88,128)( 89,129)( 90,130)( 91,131)( 92,132)( 93,133)( 94,134)( 95,135)
( 96,136)( 97,137)( 98,138)( 99,139)(100,140)(101,146)(102,147)(103,148)
(104,149)(105,150)(106,141)(107,142)(108,143)(109,144)(110,145)(111,156)
(112,157)(113,158)(114,159)(115,160)(116,151)(117,152)(118,153)(119,154)
(120,155);
s1 := Sym(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 21, 26)( 22, 27)
( 23, 28)( 24, 29)( 25, 30)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)
( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81,101)( 82,102)( 83,103)
( 84,104)( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)
( 92,117)( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)
(128,148)(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)
(136,151)(137,152)(138,153)(139,154)(140,155);
s2 := Sym(160)!(  1, 81)(  2, 85)(  3, 84)(  4, 83)(  5, 82)(  6, 86)(  7, 90)
(  8, 89)(  9, 88)( 10, 87)( 11, 91)( 12, 95)( 13, 94)( 14, 93)( 15, 92)
( 16, 96)( 17,100)( 18, 99)( 19, 98)( 20, 97)( 21,106)( 22,110)( 23,109)
( 24,108)( 25,107)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31,116)
( 32,120)( 33,119)( 34,118)( 35,117)( 36,111)( 37,115)( 38,114)( 39,113)
( 40,112)( 41,121)( 42,125)( 43,124)( 44,123)( 45,122)( 46,126)( 47,130)
( 48,129)( 49,128)( 50,127)( 51,131)( 52,135)( 53,134)( 54,133)( 55,132)
( 56,136)( 57,140)( 58,139)( 59,138)( 60,137)( 61,146)( 62,150)( 63,149)
( 64,148)( 65,147)( 66,141)( 67,145)( 68,144)( 69,143)( 70,142)( 71,156)
( 72,160)( 73,159)( 74,158)( 75,157)( 76,151)( 77,155)( 78,154)( 79,153)
( 80,152);
s3 := Sym(160)!(  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)
(  8, 46)(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)
( 16, 58)( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 68)( 22, 67)( 23, 66)
( 24, 70)( 25, 69)( 26, 63)( 27, 62)( 28, 61)( 29, 65)( 30, 64)( 31, 78)
( 32, 77)( 33, 76)( 34, 80)( 35, 79)( 36, 73)( 37, 72)( 38, 71)( 39, 75)
( 40, 74)( 81,133)( 82,132)( 83,131)( 84,135)( 85,134)( 86,138)( 87,137)
( 88,136)( 89,140)( 90,139)( 91,123)( 92,122)( 93,121)( 94,125)( 95,124)
( 96,128)( 97,127)( 98,126)( 99,130)(100,129)(101,158)(102,157)(103,156)
(104,160)(105,159)(106,153)(107,152)(108,151)(109,155)(110,154)(111,148)
(112,147)(113,146)(114,150)(115,149)(116,143)(117,142)(118,141)(119,145)
(120,144);
poly := sub<Sym(160)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope