Polytope of Type {20,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4,4}*1280a
Also Known As : {{20,4|2},{4,4}₄}. if this polytope has another name.
Group : SmallGroup(1280,201148)
Rank : 4
Schlafli Type : {20,4,4}
Number of vertices, edges, etc : 20, 80, 16, 8
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 : {20,4,4}*640, {10,4,4}*640
   4-fold quotients : {20,4,2}*320, {20,2,4}*320, {10,4,4}*320
   5-fold quotients : {4,4,4}*256a
   8-fold quotients : {20,2,2}*160, {10,2,4}*160, {10,4,2}*160
   10-fold quotients : {4,4,4}*128, {2,4,4}*128
   16-fold quotients : {5,2,4}*80, {10,2,2}*80
   20-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   32-fold quotients : {5,2,2}*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, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 56)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)( 24, 63)
( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)( 32, 75)
( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)
( 81,131)( 82,135)( 83,134)( 84,133)( 85,132)( 86,136)( 87,140)( 88,139)
( 89,138)( 90,137)( 91,121)( 92,125)( 93,124)( 94,123)( 95,122)( 96,126)
( 97,130)( 98,129)( 99,128)(100,127)(101,151)(102,155)(103,154)(104,153)
(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,141)(112,145)
(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)(120,147);;
s1 := (  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)(  8, 86)
(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)( 16, 98)
( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,103)( 22,102)( 23,101)( 24,105)
( 25,104)( 26,108)( 27,107)( 28,106)( 29,110)( 30,109)( 31,113)( 32,112)
( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)
( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)( 48,126)
( 49,130)( 50,129)( 51,133)( 52,132)( 53,131)( 54,135)( 55,134)( 56,138)
( 57,137)( 58,136)( 59,140)( 60,139)( 61,143)( 62,142)( 63,141)( 64,145)
( 65,144)( 66,148)( 67,147)( 68,146)( 69,150)( 70,149)( 71,153)( 72,152)
( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)( 80,159);;
s2 := ( 81,101)( 82,102)( 83,103)( 84,104)( 85,105)( 86,106)( 87,107)( 88,108)
( 89,109)( 90,110)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)( 96,116)
( 97,117)( 98,118)( 99,119)(100,120)(121,141)(122,142)(123,143)(124,144)
(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)(132,152)
(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)(140,160);;
s3 := (  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);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(160)!(  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)
(  8, 49)(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)
( 16, 56)( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 61)( 22, 65)( 23, 64)
( 24, 63)( 25, 62)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 71)
( 32, 75)( 33, 74)( 34, 73)( 35, 72)( 36, 76)( 37, 80)( 38, 79)( 39, 78)
( 40, 77)( 81,131)( 82,135)( 83,134)( 84,133)( 85,132)( 86,136)( 87,140)
( 88,139)( 89,138)( 90,137)( 91,121)( 92,125)( 93,124)( 94,123)( 95,122)
( 96,126)( 97,130)( 98,129)( 99,128)(100,127)(101,151)(102,155)(103,154)
(104,153)(105,152)(106,156)(107,160)(108,159)(109,158)(110,157)(111,141)
(112,145)(113,144)(114,143)(115,142)(116,146)(117,150)(118,149)(119,148)
(120,147);
s1 := Sym(160)!(  1, 83)(  2, 82)(  3, 81)(  4, 85)(  5, 84)(  6, 88)(  7, 87)
(  8, 86)(  9, 90)( 10, 89)( 11, 93)( 12, 92)( 13, 91)( 14, 95)( 15, 94)
( 16, 98)( 17, 97)( 18, 96)( 19,100)( 20, 99)( 21,103)( 22,102)( 23,101)
( 24,105)( 25,104)( 26,108)( 27,107)( 28,106)( 29,110)( 30,109)( 31,113)
( 32,112)( 33,111)( 34,115)( 35,114)( 36,118)( 37,117)( 38,116)( 39,120)
( 40,119)( 41,123)( 42,122)( 43,121)( 44,125)( 45,124)( 46,128)( 47,127)
( 48,126)( 49,130)( 50,129)( 51,133)( 52,132)( 53,131)( 54,135)( 55,134)
( 56,138)( 57,137)( 58,136)( 59,140)( 60,139)( 61,143)( 62,142)( 63,141)
( 64,145)( 65,144)( 66,148)( 67,147)( 68,146)( 69,150)( 70,149)( 71,153)
( 72,152)( 73,151)( 74,155)( 75,154)( 76,158)( 77,157)( 78,156)( 79,160)
( 80,159);
s2 := Sym(160)!( 81,101)( 82,102)( 83,103)( 84,104)( 85,105)( 86,106)( 87,107)
( 88,108)( 89,109)( 90,110)( 91,111)( 92,112)( 93,113)( 94,114)( 95,115)
( 96,116)( 97,117)( 98,118)( 99,119)(100,120)(121,141)(122,142)(123,143)
(124,144)(125,145)(126,146)(127,147)(128,148)(129,149)(130,150)(131,151)
(132,152)(133,153)(134,154)(135,155)(136,156)(137,157)(138,158)(139,159)
(140,160);
s3 := 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);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope