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Polytope of Type {3,6,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,20}*960
Also Known As : {{3,6}4,{6,20|2}}. if this polytope has another name.
Group : SmallGroup(960,10952)
Rank : 4
Schlafli Type : {3,6,20}
Number of vertices, edges, etc : 4, 12, 80, 20
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,20,2} of size 1920
Vertex Figure Of :
   {2,3,6,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,10}*480
   5-fold quotients : {3,6,4}*192
   10-fold quotients : {3,6,2}*96
   20-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12,20}*1920, {3,6,40}*1920, {6,6,20}*1920
Permutation Representation (GAP) :
s0 := (  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)( 23, 42)
( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)( 31, 50)
( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)( 39, 58)
( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)( 82,103)
( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)( 90,111)
( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)( 98,119)
( 99,118)(100,120)(122,123)(126,127)(130,131)(134,135)(138,139)(141,161)
(142,163)(143,162)(144,164)(145,165)(146,167)(147,166)(148,168)(149,169)
(150,171)(151,170)(152,172)(153,173)(154,175)(155,174)(156,176)(157,177)
(158,179)(159,178)(160,180)(182,183)(186,187)(190,191)(194,195)(198,199)
(201,221)(202,223)(203,222)(204,224)(205,225)(206,227)(207,226)(208,228)
(209,229)(210,231)(211,230)(212,232)(213,233)(214,235)(215,234)(216,236)
(217,237)(218,239)(219,238)(220,240);;
s1 := (  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 25)(  6, 26)(  7, 28)(  8, 27)
(  9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)( 16, 35)
( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)( 55, 56)
( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)( 67, 88)
( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)(111,112)
(115,116)(119,120)(121,141)(122,142)(123,144)(124,143)(125,145)(126,146)
(127,148)(128,147)(129,149)(130,150)(131,152)(132,151)(133,153)(134,154)
(135,156)(136,155)(137,157)(138,158)(139,160)(140,159)(163,164)(167,168)
(171,172)(175,176)(179,180)(181,201)(182,202)(183,204)(184,203)(185,205)
(186,206)(187,208)(188,207)(189,209)(190,210)(191,212)(192,211)(193,213)
(194,214)(195,216)(196,215)(197,217)(198,218)(199,220)(200,219)(223,224)
(227,228)(231,232)(235,236)(239,240);;
s2 := (  1,  4)(  5, 20)(  6, 18)(  7, 19)(  8, 17)(  9, 16)( 10, 14)( 11, 15)
( 12, 13)( 21, 44)( 22, 42)( 23, 43)( 24, 41)( 25, 60)( 26, 58)( 27, 59)
( 28, 57)( 29, 56)( 30, 54)( 31, 55)( 32, 53)( 33, 52)( 34, 50)( 35, 51)
( 36, 49)( 37, 48)( 38, 46)( 39, 47)( 40, 45)( 61, 64)( 65, 80)( 66, 78)
( 67, 79)( 68, 77)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 81,104)( 82,102)
( 83,103)( 84,101)( 85,120)( 86,118)( 87,119)( 88,117)( 89,116)( 90,114)
( 91,115)( 92,113)( 93,112)( 94,110)( 95,111)( 96,109)( 97,108)( 98,106)
( 99,107)(100,105)(121,184)(122,182)(123,183)(124,181)(125,200)(126,198)
(127,199)(128,197)(129,196)(130,194)(131,195)(132,193)(133,192)(134,190)
(135,191)(136,189)(137,188)(138,186)(139,187)(140,185)(141,224)(142,222)
(143,223)(144,221)(145,240)(146,238)(147,239)(148,237)(149,236)(150,234)
(151,235)(152,233)(153,232)(154,230)(155,231)(156,229)(157,228)(158,226)
(159,227)(160,225)(161,204)(162,202)(163,203)(164,201)(165,220)(166,218)
(167,219)(168,217)(169,216)(170,214)(171,215)(172,213)(173,212)(174,210)
(175,211)(176,209)(177,208)(178,206)(179,207)(180,205);;
s3 := (  1,125)(  2,126)(  3,127)(  4,128)(  5,121)(  6,122)(  7,123)(  8,124)
(  9,137)( 10,138)( 11,139)( 12,140)( 13,133)( 14,134)( 15,135)( 16,136)
( 17,129)( 18,130)( 19,131)( 20,132)( 21,145)( 22,146)( 23,147)( 24,148)
( 25,141)( 26,142)( 27,143)( 28,144)( 29,157)( 30,158)( 31,159)( 32,160)
( 33,153)( 34,154)( 35,155)( 36,156)( 37,149)( 38,150)( 39,151)( 40,152)
( 41,165)( 42,166)( 43,167)( 44,168)( 45,161)( 46,162)( 47,163)( 48,164)
( 49,177)( 50,178)( 51,179)( 52,180)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,169)( 58,170)( 59,171)( 60,172)( 61,185)( 62,186)( 63,187)( 64,188)
( 65,181)( 66,182)( 67,183)( 68,184)( 69,197)( 70,198)( 71,199)( 72,200)
( 73,193)( 74,194)( 75,195)( 76,196)( 77,189)( 78,190)( 79,191)( 80,192)
( 81,205)( 82,206)( 83,207)( 84,208)( 85,201)( 86,202)( 87,203)( 88,204)
( 89,217)( 90,218)( 91,219)( 92,220)( 93,213)( 94,214)( 95,215)( 96,216)
( 97,209)( 98,210)( 99,211)(100,212)(101,225)(102,226)(103,227)(104,228)
(105,221)(106,222)(107,223)(108,224)(109,237)(110,238)(111,239)(112,240)
(113,233)(114,234)(115,235)(116,236)(117,229)(118,230)(119,231)(120,232);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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(240)!(  2,  3)(  6,  7)( 10, 11)( 14, 15)( 18, 19)( 21, 41)( 22, 43)
( 23, 42)( 24, 44)( 25, 45)( 26, 47)( 27, 46)( 28, 48)( 29, 49)( 30, 51)
( 31, 50)( 32, 52)( 33, 53)( 34, 55)( 35, 54)( 36, 56)( 37, 57)( 38, 59)
( 39, 58)( 40, 60)( 62, 63)( 66, 67)( 70, 71)( 74, 75)( 78, 79)( 81,101)
( 82,103)( 83,102)( 84,104)( 85,105)( 86,107)( 87,106)( 88,108)( 89,109)
( 90,111)( 91,110)( 92,112)( 93,113)( 94,115)( 95,114)( 96,116)( 97,117)
( 98,119)( 99,118)(100,120)(122,123)(126,127)(130,131)(134,135)(138,139)
(141,161)(142,163)(143,162)(144,164)(145,165)(146,167)(147,166)(148,168)
(149,169)(150,171)(151,170)(152,172)(153,173)(154,175)(155,174)(156,176)
(157,177)(158,179)(159,178)(160,180)(182,183)(186,187)(190,191)(194,195)
(198,199)(201,221)(202,223)(203,222)(204,224)(205,225)(206,227)(207,226)
(208,228)(209,229)(210,231)(211,230)(212,232)(213,233)(214,235)(215,234)
(216,236)(217,237)(218,239)(219,238)(220,240);
s1 := Sym(240)!(  1, 21)(  2, 22)(  3, 24)(  4, 23)(  5, 25)(  6, 26)(  7, 28)
(  8, 27)(  9, 29)( 10, 30)( 11, 32)( 12, 31)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 37)( 18, 38)( 19, 40)( 20, 39)( 43, 44)( 47, 48)( 51, 52)
( 55, 56)( 59, 60)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 85)( 66, 86)
( 67, 88)( 68, 87)( 69, 89)( 70, 90)( 71, 92)( 72, 91)( 73, 93)( 74, 94)
( 75, 96)( 76, 95)( 77, 97)( 78, 98)( 79,100)( 80, 99)(103,104)(107,108)
(111,112)(115,116)(119,120)(121,141)(122,142)(123,144)(124,143)(125,145)
(126,146)(127,148)(128,147)(129,149)(130,150)(131,152)(132,151)(133,153)
(134,154)(135,156)(136,155)(137,157)(138,158)(139,160)(140,159)(163,164)
(167,168)(171,172)(175,176)(179,180)(181,201)(182,202)(183,204)(184,203)
(185,205)(186,206)(187,208)(188,207)(189,209)(190,210)(191,212)(192,211)
(193,213)(194,214)(195,216)(196,215)(197,217)(198,218)(199,220)(200,219)
(223,224)(227,228)(231,232)(235,236)(239,240);
s2 := Sym(240)!(  1,  4)(  5, 20)(  6, 18)(  7, 19)(  8, 17)(  9, 16)( 10, 14)
( 11, 15)( 12, 13)( 21, 44)( 22, 42)( 23, 43)( 24, 41)( 25, 60)( 26, 58)
( 27, 59)( 28, 57)( 29, 56)( 30, 54)( 31, 55)( 32, 53)( 33, 52)( 34, 50)
( 35, 51)( 36, 49)( 37, 48)( 38, 46)( 39, 47)( 40, 45)( 61, 64)( 65, 80)
( 66, 78)( 67, 79)( 68, 77)( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 81,104)
( 82,102)( 83,103)( 84,101)( 85,120)( 86,118)( 87,119)( 88,117)( 89,116)
( 90,114)( 91,115)( 92,113)( 93,112)( 94,110)( 95,111)( 96,109)( 97,108)
( 98,106)( 99,107)(100,105)(121,184)(122,182)(123,183)(124,181)(125,200)
(126,198)(127,199)(128,197)(129,196)(130,194)(131,195)(132,193)(133,192)
(134,190)(135,191)(136,189)(137,188)(138,186)(139,187)(140,185)(141,224)
(142,222)(143,223)(144,221)(145,240)(146,238)(147,239)(148,237)(149,236)
(150,234)(151,235)(152,233)(153,232)(154,230)(155,231)(156,229)(157,228)
(158,226)(159,227)(160,225)(161,204)(162,202)(163,203)(164,201)(165,220)
(166,218)(167,219)(168,217)(169,216)(170,214)(171,215)(172,213)(173,212)
(174,210)(175,211)(176,209)(177,208)(178,206)(179,207)(180,205);
s3 := Sym(240)!(  1,125)(  2,126)(  3,127)(  4,128)(  5,121)(  6,122)(  7,123)
(  8,124)(  9,137)( 10,138)( 11,139)( 12,140)( 13,133)( 14,134)( 15,135)
( 16,136)( 17,129)( 18,130)( 19,131)( 20,132)( 21,145)( 22,146)( 23,147)
( 24,148)( 25,141)( 26,142)( 27,143)( 28,144)( 29,157)( 30,158)( 31,159)
( 32,160)( 33,153)( 34,154)( 35,155)( 36,156)( 37,149)( 38,150)( 39,151)
( 40,152)( 41,165)( 42,166)( 43,167)( 44,168)( 45,161)( 46,162)( 47,163)
( 48,164)( 49,177)( 50,178)( 51,179)( 52,180)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,169)( 58,170)( 59,171)( 60,172)( 61,185)( 62,186)( 63,187)
( 64,188)( 65,181)( 66,182)( 67,183)( 68,184)( 69,197)( 70,198)( 71,199)
( 72,200)( 73,193)( 74,194)( 75,195)( 76,196)( 77,189)( 78,190)( 79,191)
( 80,192)( 81,205)( 82,206)( 83,207)( 84,208)( 85,201)( 86,202)( 87,203)
( 88,204)( 89,217)( 90,218)( 91,219)( 92,220)( 93,213)( 94,214)( 95,215)
( 96,216)( 97,209)( 98,210)( 99,211)(100,212)(101,225)(102,226)(103,227)
(104,228)(105,221)(106,222)(107,223)(108,224)(109,237)(110,238)(111,239)
(112,240)(113,233)(114,234)(115,235)(116,236)(117,229)(118,230)(119,231)
(120,232);
poly := sub<Sym(240)|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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
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