Polytope of Type {6,20,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,20,4}*1440
if this polytope has a name.
Group : SmallGroup(1440,5199)
Rank : 4
Schlafli Type : {6,20,4}
Number of vertices, edges, etc : 9, 90, 60, 4
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 : {6,20,2}*720
   5-fold quotients : {6,4,4}*288
   10-fold quotients : {6,4,2}*144
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope