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Polytope of Type {6,4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,20}*1920a
if this polytope has a name.
Group : SmallGroup(1920,151308)
Rank : 4
Schlafli Type : {6,4,20}
Number of vertices, edges, etc : 6, 24, 80, 40
Order of s0s1s2s3 : 60
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 : {6,4,20}*960
   3-fold quotients : {2,4,20}*640
   4-fold quotients : {6,2,20}*480, {6,4,10}*480
   5-fold quotients : {6,4,4}*384a
   6-fold quotients : {2,4,20}*320
   8-fold quotients : {3,2,20}*240, {6,2,10}*240
   10-fold quotients : {6,4,4}*192
   12-fold quotients : {2,2,20}*160, {2,4,10}*160
   15-fold quotients : {2,4,4}*128
   16-fold quotients : {3,2,10}*120, {6,2,5}*120
   20-fold quotients : {6,2,4}*96, {6,4,2}*96a
   24-fold quotients : {2,2,10}*80
   30-fold quotients : {2,4,4}*64
   32-fold quotients : {3,2,5}*60
   40-fold quotients : {3,2,4}*48, {6,2,2}*48
   48-fold quotients : {2,2,5}*40
   60-fold quotients : {2,2,4}*32, {2,4,2}*32
   80-fold quotients : {3,2,2}*24
   120-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)(  8, 73)
(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 76)
( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31, 91)( 32, 92)
( 33, 93)( 34, 94)( 35, 95)( 36,101)( 37,102)( 38,103)( 39,104)( 40,105)
( 41, 96)( 42, 97)( 43, 98)( 44, 99)( 45,100)( 46,106)( 47,107)( 48,108)
( 49,109)( 50,110)( 51,116)( 52,117)( 53,118)( 54,119)( 55,120)( 56,111)
( 57,112)( 58,113)( 59,114)( 60,115)(121,181)(122,182)(123,183)(124,184)
(125,185)(126,191)(127,192)(128,193)(129,194)(130,195)(131,186)(132,187)
(133,188)(134,189)(135,190)(136,196)(137,197)(138,198)(139,199)(140,200)
(141,206)(142,207)(143,208)(144,209)(145,210)(146,201)(147,202)(148,203)
(149,204)(150,205)(151,211)(152,212)(153,213)(154,214)(155,215)(156,221)
(157,222)(158,223)(159,224)(160,225)(161,216)(162,217)(163,218)(164,219)
(165,220)(166,226)(167,227)(168,228)(169,229)(170,230)(171,236)(172,237)
(173,238)(174,239)(175,240)(176,231)(177,232)(178,233)(179,234)(180,235);;
s1 := (  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)(  8, 63)
(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)( 16, 81)
( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)( 24, 79)
( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31,111)( 32,112)
( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)( 40,110)
( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46, 96)( 47, 97)( 48, 98)
( 49, 99)( 50,100)( 51, 91)( 52, 92)( 53, 93)( 54, 94)( 55, 95)( 56,101)
( 57,102)( 58,103)( 59,104)( 60,105)(121,186)(122,187)(123,188)(124,189)
(125,190)(126,181)(127,182)(128,183)(129,184)(130,185)(131,191)(132,192)
(133,193)(134,194)(135,195)(136,201)(137,202)(138,203)(139,204)(140,205)
(141,196)(142,197)(143,198)(144,199)(145,200)(146,206)(147,207)(148,208)
(149,209)(150,210)(151,231)(152,232)(153,233)(154,234)(155,235)(156,226)
(157,227)(158,228)(159,229)(160,230)(161,236)(162,237)(163,238)(164,239)
(165,240)(166,216)(167,217)(168,218)(169,219)(170,220)(171,211)(172,212)
(173,213)(174,214)(175,215)(176,221)(177,222)(178,223)(179,224)(180,225);;
s2 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99)
(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)(118,119)
(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)(127,160)(128,159)
(129,158)(130,157)(131,161)(132,165)(133,164)(134,163)(135,162)(136,166)
(137,170)(138,169)(139,168)(140,167)(141,171)(142,175)(143,174)(144,173)
(145,172)(146,176)(147,180)(148,179)(149,178)(150,177)(181,211)(182,215)
(183,214)(184,213)(185,212)(186,216)(187,220)(188,219)(189,218)(190,217)
(191,221)(192,225)(193,224)(194,223)(195,222)(196,226)(197,230)(198,229)
(199,228)(200,227)(201,231)(202,235)(203,234)(204,233)(205,232)(206,236)
(207,240)(208,239)(209,238)(210,237);;
s3 := (  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)(  8,130)
(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)( 16,137)
( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)( 24,144)
( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,152)( 32,151)
( 33,155)( 34,154)( 35,153)( 36,157)( 37,156)( 38,160)( 39,159)( 40,158)
( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)( 48,170)
( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)( 56,177)
( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)( 64,184)
( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)( 72,191)
( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)( 80,198)
( 81,202)( 82,201)( 83,205)( 84,204)( 85,203)( 86,207)( 87,206)( 88,210)
( 89,209)( 90,208)( 91,212)( 92,211)( 93,215)( 94,214)( 95,213)( 96,217)
( 97,216)( 98,220)( 99,219)(100,218)(101,222)(102,221)(103,225)(104,224)
(105,223)(106,227)(107,226)(108,230)(109,229)(110,228)(111,232)(112,231)
(113,235)(114,234)(115,233)(116,237)(117,236)(118,240)(119,239)(120,238);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*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(240)!(  1, 61)(  2, 62)(  3, 63)(  4, 64)(  5, 65)(  6, 71)(  7, 72)
(  8, 73)(  9, 74)( 10, 75)( 11, 66)( 12, 67)( 13, 68)( 14, 69)( 15, 70)
( 16, 76)( 17, 77)( 18, 78)( 19, 79)( 20, 80)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26, 81)( 27, 82)( 28, 83)( 29, 84)( 30, 85)( 31, 91)
( 32, 92)( 33, 93)( 34, 94)( 35, 95)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,105)( 41, 96)( 42, 97)( 43, 98)( 44, 99)( 45,100)( 46,106)( 47,107)
( 48,108)( 49,109)( 50,110)( 51,116)( 52,117)( 53,118)( 54,119)( 55,120)
( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)(121,181)(122,182)(123,183)
(124,184)(125,185)(126,191)(127,192)(128,193)(129,194)(130,195)(131,186)
(132,187)(133,188)(134,189)(135,190)(136,196)(137,197)(138,198)(139,199)
(140,200)(141,206)(142,207)(143,208)(144,209)(145,210)(146,201)(147,202)
(148,203)(149,204)(150,205)(151,211)(152,212)(153,213)(154,214)(155,215)
(156,221)(157,222)(158,223)(159,224)(160,225)(161,216)(162,217)(163,218)
(164,219)(165,220)(166,226)(167,227)(168,228)(169,229)(170,230)(171,236)
(172,237)(173,238)(174,239)(175,240)(176,231)(177,232)(178,233)(179,234)
(180,235);
s1 := Sym(240)!(  1, 66)(  2, 67)(  3, 68)(  4, 69)(  5, 70)(  6, 61)(  7, 62)
(  8, 63)(  9, 64)( 10, 65)( 11, 71)( 12, 72)( 13, 73)( 14, 74)( 15, 75)
( 16, 81)( 17, 82)( 18, 83)( 19, 84)( 20, 85)( 21, 76)( 22, 77)( 23, 78)
( 24, 79)( 25, 80)( 26, 86)( 27, 87)( 28, 88)( 29, 89)( 30, 90)( 31,111)
( 32,112)( 33,113)( 34,114)( 35,115)( 36,106)( 37,107)( 38,108)( 39,109)
( 40,110)( 41,116)( 42,117)( 43,118)( 44,119)( 45,120)( 46, 96)( 47, 97)
( 48, 98)( 49, 99)( 50,100)( 51, 91)( 52, 92)( 53, 93)( 54, 94)( 55, 95)
( 56,101)( 57,102)( 58,103)( 59,104)( 60,105)(121,186)(122,187)(123,188)
(124,189)(125,190)(126,181)(127,182)(128,183)(129,184)(130,185)(131,191)
(132,192)(133,193)(134,194)(135,195)(136,201)(137,202)(138,203)(139,204)
(140,205)(141,196)(142,197)(143,198)(144,199)(145,200)(146,206)(147,207)
(148,208)(149,209)(150,210)(151,231)(152,232)(153,233)(154,234)(155,235)
(156,226)(157,227)(158,228)(159,229)(160,230)(161,236)(162,237)(163,238)
(164,239)(165,240)(166,216)(167,217)(168,218)(169,219)(170,220)(171,211)
(172,212)(173,213)(174,214)(175,215)(176,221)(177,222)(178,223)(179,224)
(180,225);
s2 := Sym(240)!(  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)
( 98, 99)(102,105)(103,104)(107,110)(108,109)(112,115)(113,114)(117,120)
(118,119)(121,151)(122,155)(123,154)(124,153)(125,152)(126,156)(127,160)
(128,159)(129,158)(130,157)(131,161)(132,165)(133,164)(134,163)(135,162)
(136,166)(137,170)(138,169)(139,168)(140,167)(141,171)(142,175)(143,174)
(144,173)(145,172)(146,176)(147,180)(148,179)(149,178)(150,177)(181,211)
(182,215)(183,214)(184,213)(185,212)(186,216)(187,220)(188,219)(189,218)
(190,217)(191,221)(192,225)(193,224)(194,223)(195,222)(196,226)(197,230)
(198,229)(199,228)(200,227)(201,231)(202,235)(203,234)(204,233)(205,232)
(206,236)(207,240)(208,239)(209,238)(210,237);
s3 := Sym(240)!(  1,122)(  2,121)(  3,125)(  4,124)(  5,123)(  6,127)(  7,126)
(  8,130)(  9,129)( 10,128)( 11,132)( 12,131)( 13,135)( 14,134)( 15,133)
( 16,137)( 17,136)( 18,140)( 19,139)( 20,138)( 21,142)( 22,141)( 23,145)
( 24,144)( 25,143)( 26,147)( 27,146)( 28,150)( 29,149)( 30,148)( 31,152)
( 32,151)( 33,155)( 34,154)( 35,153)( 36,157)( 37,156)( 38,160)( 39,159)
( 40,158)( 41,162)( 42,161)( 43,165)( 44,164)( 45,163)( 46,167)( 47,166)
( 48,170)( 49,169)( 50,168)( 51,172)( 52,171)( 53,175)( 54,174)( 55,173)
( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,182)( 62,181)( 63,185)
( 64,184)( 65,183)( 66,187)( 67,186)( 68,190)( 69,189)( 70,188)( 71,192)
( 72,191)( 73,195)( 74,194)( 75,193)( 76,197)( 77,196)( 78,200)( 79,199)
( 80,198)( 81,202)( 82,201)( 83,205)( 84,204)( 85,203)( 86,207)( 87,206)
( 88,210)( 89,209)( 90,208)( 91,212)( 92,211)( 93,215)( 94,214)( 95,213)
( 96,217)( 97,216)( 98,220)( 99,219)(100,218)(101,222)(102,221)(103,225)
(104,224)(105,223)(106,227)(107,226)(108,230)(109,229)(110,228)(111,232)
(112,231)(113,235)(114,234)(115,233)(116,237)(117,236)(118,240)(119,239)
(120,238);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*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