Polytope of Type {4,20,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20,6}*1440
if this polytope has a name.
Group : SmallGroup(1440,5199)
Rank : 4
Schlafli Type : {4,20,6}
Number of vertices, edges, etc : 4, 60, 90, 9
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-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 : {2,20,6}*720
   5-fold quotients : {4,4,6}*288
   10-fold quotients : {2,4,6}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) :

s0 := (  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)( 89,179)( 90,180);;
s1 := (  2,  5)(  3,  4)(  6, 31)(  7, 35)(  8, 34)(  9, 33)( 10, 32)( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 27, 30)( 28, 29)( 37, 40)( 38, 39)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 61)( 57, 65)( 58, 64)( 59, 63)( 60, 62)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)( 96,166)( 97,170)( 98,169)( 99,168)(100,167)(101,151)(102,155)(103,154)(104,153)(105,152)(106,146)(107,150)(108,149)(109,148)(110,147)(111,176)(112,180)(113,179)(114,178)(115,177)(116,161)(117,165)(118,164)(119,163)(120,162)(121,141)(122,145)(123,144)(124,143)(125,142)(126,171)(127,175)(128,174)(129,173)(130,172)(131,156)(132,160)(133,159)(134,158)(135,157);;
s2 := (  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)(113,130)(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)(136,137)(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)(153,170)(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)(161,177)(162,176)(163,180)(164,179)(165,178);;
s3 := (  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 31)(  7, 32)(  8, 33)(  9, 34)( 10, 35)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 21)( 17, 22)( 18, 23)( 19, 24)( 20, 25)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)( 51, 76)( 52, 77)( 53, 78)( 54, 79)( 55, 80)( 56, 86)( 57, 87)( 58, 88)( 59, 89)( 60, 90)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 91,126)( 92,127)( 93,128)( 94,129)( 95,130)( 96,121)( 97,122)( 98,123)( 99,124)(100,125)(101,131)(102,132)(103,133)(104,134)(105,135)(106,111)(107,112)(108,113)(109,114)(110,115)(136,171)(137,172)(138,173)(139,174)(140,175)(141,166)(142,167)(143,168)(144,169)(145,170)(146,176)(147,177)(148,178)(149,179)(150,180)(151,156)(152,157)(153,158)(154,159)(155,160);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(180)!(  1, 91)(  2, 92)(  3, 93)(  4, 94)(  5, 95)(  6, 96)(  7, 97)(  8, 98)(  9, 99)( 10,100)( 11,101)( 12,102)( 13,103)( 14,104)( 15,105)( 16,106)( 17,107)( 18,108)( 19,109)( 20,110)( 21,111)( 22,112)( 23,113)( 24,114)( 25,115)( 26,116)( 27,117)( 28,118)( 29,119)( 30,120)( 31,121)( 32,122)( 33,123)( 34,124)( 35,125)( 36,126)( 37,127)( 38,128)( 39,129)( 40,130)( 41,131)( 42,132)( 43,133)( 44,134)( 45,135)( 46,136)( 47,137)( 48,138)( 49,139)( 50,140)( 51,141)( 52,142)( 53,143)( 54,144)( 55,145)( 56,146)( 57,147)( 58,148)( 59,149)( 60,150)( 61,151)( 62,152)( 63,153)( 64,154)( 65,155)( 66,156)( 67,157)( 68,158)( 69,159)( 70,160)( 71,161)( 72,162)( 73,163)( 74,164)( 75,165)( 76,166)( 77,167)( 78,168)( 79,169)( 80,170)( 81,171)( 82,172)( 83,173)( 84,174)( 85,175)( 86,176)( 87,177)( 88,178)( 89,179)( 90,180);
s1 := Sym(180)!(  2,  5)(  3,  4)(  6, 31)(  7, 35)(  8, 34)(  9, 33)( 10, 32)( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 21, 41)( 22, 45)( 23, 44)( 24, 43)( 25, 42)( 27, 30)( 28, 29)( 37, 40)( 38, 39)( 47, 50)( 48, 49)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 61)( 57, 65)( 58, 64)( 59, 63)( 60, 62)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 72, 75)( 73, 74)( 82, 85)( 83, 84)( 91,136)( 92,140)( 93,139)( 94,138)( 95,137)( 96,166)( 97,170)( 98,169)( 99,168)(100,167)(101,151)(102,155)(103,154)(104,153)(105,152)(106,146)(107,150)(108,149)(109,148)(110,147)(111,176)(112,180)(113,179)(114,178)(115,177)(116,161)(117,165)(118,164)(119,163)(120,162)(121,141)(122,145)(123,144)(124,143)(125,142)(126,171)(127,175)(128,174)(129,173)(130,172)(131,156)(132,160)(133,159)(134,158)(135,157);
s2 := Sym(180)!(  1,  2)(  3,  5)(  6,  7)(  8, 10)( 11, 12)( 13, 15)( 16, 32)( 17, 31)( 18, 35)( 19, 34)( 20, 33)( 21, 37)( 22, 36)( 23, 40)( 24, 39)( 25, 38)( 26, 42)( 27, 41)( 28, 45)( 29, 44)( 30, 43)( 46, 47)( 48, 50)( 51, 52)( 53, 55)( 56, 57)( 58, 60)( 61, 77)( 62, 76)( 63, 80)( 64, 79)( 65, 78)( 66, 82)( 67, 81)( 68, 85)( 69, 84)( 70, 83)( 71, 87)( 72, 86)( 73, 90)( 74, 89)( 75, 88)( 91, 92)( 93, 95)( 96, 97)( 98,100)(101,102)(103,105)(106,122)(107,121)(108,125)(109,124)(110,123)(111,127)(112,126)(113,130)(114,129)(115,128)(116,132)(117,131)(118,135)(119,134)(120,133)(136,137)(138,140)(141,142)(143,145)(146,147)(148,150)(151,167)(152,166)(153,170)(154,169)(155,168)(156,172)(157,171)(158,175)(159,174)(160,173)(161,177)(162,176)(163,180)(164,179)(165,178);
s3 := Sym(180)!(  1, 36)(  2, 37)(  3, 38)(  4, 39)(  5, 40)(  6, 31)(  7, 32)(  8, 33)(  9, 34)( 10, 35)( 11, 41)( 12, 42)( 13, 43)( 14, 44)( 15, 45)( 16, 21)( 17, 22)( 18, 23)( 19, 24)( 20, 25)( 46, 81)( 47, 82)( 48, 83)( 49, 84)( 50, 85)( 51, 76)( 52, 77)( 53, 78)( 54, 79)( 55, 80)( 56, 86)( 57, 87)( 58, 88)( 59, 89)( 60, 90)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 91,126)( 92,127)( 93,128)( 94,129)( 95,130)( 96,121)( 97,122)( 98,123)( 99,124)(100,125)(101,131)(102,132)(103,133)(104,134)(105,135)(106,111)(107,112)(108,113)(109,114)(110,115)(136,171)(137,172)(138,173)(139,174)(140,175)(141,166)(142,167)(143,168)(144,169)(145,170)(146,176)(147,177)(148,178)(149,179)(150,180)(151,156)(152,157)(153,158)(154,159)(155,160);
poly := sub<Sym(180)|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, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s3*s2*s1*s2*s1*s3*s2*s1*s2*s3 >;

References : None.
to this polytope