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Polytope of Type {3,6,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4,10}*1920
Also Known As : {{3,6}4,{6,4|2},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(1920,240151)
Rank : 5
Schlafli Type : {3,6,4,10}
Number of vertices, edges, etc : 4, 12, 16, 20, 10
Order of s0s1s2s3s4 : 20
Order of s0s1s2s3s4s3s2s1 : 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 : {3,6,2,10}*960
   4-fold quotients : {3,3,2,10}*480, {3,6,2,5}*480
   5-fold quotients : {3,6,4,2}*384
   8-fold quotients : {3,3,2,5}*240
   10-fold quotients : {3,6,2,2}*192
   20-fold quotients : {3,3,2,2}*96
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)( 23, 44)
( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)( 31, 52)
( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)( 39, 60)
( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)( 82,102)
( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)( 98,118)
( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)(141,161)
(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)(149,169)
(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)(157,177)
(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)(199,200)
(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)(208,227)
(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)(216,235)
(217,237)(218,238)(219,240)(220,239);;
s1 := (  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)(  8, 26)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)( 16, 34)
( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)( 54, 56)
( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)( 67, 87)
( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)( 75, 95)
( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)(110,112)
(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)(126,148)
(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)(134,156)
(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)(166,168)
(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)(185,205)
(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)(193,213)
(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)(222,224)
(226,228)(230,232)(234,236)(238,240);;
s2 := (  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)( 23, 43)
( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)( 31, 51)
( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)( 39, 59)
( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)( 82,101)
( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)( 90,109)
( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)( 98,117)
( 99,119)(100,120)(121,182)(122,181)(123,183)(124,184)(125,186)(126,185)
(127,187)(128,188)(129,190)(130,189)(131,191)(132,192)(133,194)(134,193)
(135,195)(136,196)(137,198)(138,197)(139,199)(140,200)(141,222)(142,221)
(143,223)(144,224)(145,226)(146,225)(147,227)(148,228)(149,230)(150,229)
(151,231)(152,232)(153,234)(154,233)(155,235)(156,236)(157,238)(158,237)
(159,239)(160,240)(161,202)(162,201)(163,203)(164,204)(165,206)(166,205)
(167,207)(168,208)(169,210)(170,209)(171,211)(172,212)(173,214)(174,213)
(175,215)(176,216)(177,218)(178,217)(179,219)(180,220);;
s3 := (  1,121)(  2,122)(  3,123)(  4,124)(  5,137)(  6,138)(  7,139)(  8,140)
(  9,133)( 10,134)( 11,135)( 12,136)( 13,129)( 14,130)( 15,131)( 16,132)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,141)( 22,142)( 23,143)( 24,144)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,153)( 30,154)( 31,155)( 32,156)
( 33,149)( 34,150)( 35,151)( 36,152)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,161)( 42,162)( 43,163)( 44,164)( 45,177)( 46,178)( 47,179)( 48,180)
( 49,173)( 50,174)( 51,175)( 52,176)( 53,169)( 54,170)( 55,171)( 56,172)
( 57,165)( 58,166)( 59,167)( 60,168)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,197)( 66,198)( 67,199)( 68,200)( 69,193)( 70,194)( 71,195)( 72,196)
( 73,189)( 74,190)( 75,191)( 76,192)( 77,185)( 78,186)( 79,187)( 80,188)
( 81,201)( 82,202)( 83,203)( 84,204)( 85,217)( 86,218)( 87,219)( 88,220)
( 89,213)( 90,214)( 91,215)( 92,216)( 93,209)( 94,210)( 95,211)( 96,212)
( 97,205)( 98,206)( 99,207)(100,208)(101,221)(102,222)(103,223)(104,224)
(105,237)(106,238)(107,239)(108,240)(109,233)(110,234)(111,235)(112,236)
(113,229)(114,230)(115,231)(116,232)(117,225)(118,226)(119,227)(120,228);;
s4 := (  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 17)( 10, 18)( 11, 19)( 12, 20)
( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)( 32, 40)
( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)
( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)( 92,100)
(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)(112,120)
(121,125)(122,126)(123,127)(124,128)(129,137)(130,138)(131,139)(132,140)
(141,145)(142,146)(143,147)(144,148)(149,157)(150,158)(151,159)(152,160)
(161,165)(162,166)(163,167)(164,168)(169,177)(170,178)(171,179)(172,180)
(181,185)(182,186)(183,187)(184,188)(189,197)(190,198)(191,199)(192,200)
(201,205)(202,206)(203,207)(204,208)(209,217)(210,218)(211,219)(212,220)
(221,225)(222,226)(223,227)(224,228)(229,237)(230,238)(231,239)(232,240);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(240)!(  3,  4)(  7,  8)( 11, 12)( 15, 16)( 19, 20)( 21, 41)( 22, 42)
( 23, 44)( 24, 43)( 25, 45)( 26, 46)( 27, 48)( 28, 47)( 29, 49)( 30, 50)
( 31, 52)( 32, 51)( 33, 53)( 34, 54)( 35, 56)( 36, 55)( 37, 57)( 38, 58)
( 39, 60)( 40, 59)( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 79, 80)( 81,101)
( 82,102)( 83,104)( 84,103)( 85,105)( 86,106)( 87,108)( 88,107)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,113)( 94,114)( 95,116)( 96,115)( 97,117)
( 98,118)( 99,120)(100,119)(123,124)(127,128)(131,132)(135,136)(139,140)
(141,161)(142,162)(143,164)(144,163)(145,165)(146,166)(147,168)(148,167)
(149,169)(150,170)(151,172)(152,171)(153,173)(154,174)(155,176)(156,175)
(157,177)(158,178)(159,180)(160,179)(183,184)(187,188)(191,192)(195,196)
(199,200)(201,221)(202,222)(203,224)(204,223)(205,225)(206,226)(207,228)
(208,227)(209,229)(210,230)(211,232)(212,231)(213,233)(214,234)(215,236)
(216,235)(217,237)(218,238)(219,240)(220,239);
s1 := Sym(240)!(  1, 21)(  2, 24)(  3, 23)(  4, 22)(  5, 25)(  6, 28)(  7, 27)
(  8, 26)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 13, 33)( 14, 36)( 15, 35)
( 16, 34)( 17, 37)( 18, 40)( 19, 39)( 20, 38)( 42, 44)( 46, 48)( 50, 52)
( 54, 56)( 58, 60)( 61, 81)( 62, 84)( 63, 83)( 64, 82)( 65, 85)( 66, 88)
( 67, 87)( 68, 86)( 69, 89)( 70, 92)( 71, 91)( 72, 90)( 73, 93)( 74, 96)
( 75, 95)( 76, 94)( 77, 97)( 78,100)( 79, 99)( 80, 98)(102,104)(106,108)
(110,112)(114,116)(118,120)(121,141)(122,144)(123,143)(124,142)(125,145)
(126,148)(127,147)(128,146)(129,149)(130,152)(131,151)(132,150)(133,153)
(134,156)(135,155)(136,154)(137,157)(138,160)(139,159)(140,158)(162,164)
(166,168)(170,172)(174,176)(178,180)(181,201)(182,204)(183,203)(184,202)
(185,205)(186,208)(187,207)(188,206)(189,209)(190,212)(191,211)(192,210)
(193,213)(194,216)(195,215)(196,214)(197,217)(198,220)(199,219)(200,218)
(222,224)(226,228)(230,232)(234,236)(238,240);
s2 := Sym(240)!(  1,  2)(  5,  6)(  9, 10)( 13, 14)( 17, 18)( 21, 42)( 22, 41)
( 23, 43)( 24, 44)( 25, 46)( 26, 45)( 27, 47)( 28, 48)( 29, 50)( 30, 49)
( 31, 51)( 32, 52)( 33, 54)( 34, 53)( 35, 55)( 36, 56)( 37, 58)( 38, 57)
( 39, 59)( 40, 60)( 61, 62)( 65, 66)( 69, 70)( 73, 74)( 77, 78)( 81,102)
( 82,101)( 83,103)( 84,104)( 85,106)( 86,105)( 87,107)( 88,108)( 89,110)
( 90,109)( 91,111)( 92,112)( 93,114)( 94,113)( 95,115)( 96,116)( 97,118)
( 98,117)( 99,119)(100,120)(121,182)(122,181)(123,183)(124,184)(125,186)
(126,185)(127,187)(128,188)(129,190)(130,189)(131,191)(132,192)(133,194)
(134,193)(135,195)(136,196)(137,198)(138,197)(139,199)(140,200)(141,222)
(142,221)(143,223)(144,224)(145,226)(146,225)(147,227)(148,228)(149,230)
(150,229)(151,231)(152,232)(153,234)(154,233)(155,235)(156,236)(157,238)
(158,237)(159,239)(160,240)(161,202)(162,201)(163,203)(164,204)(165,206)
(166,205)(167,207)(168,208)(169,210)(170,209)(171,211)(172,212)(173,214)
(174,213)(175,215)(176,216)(177,218)(178,217)(179,219)(180,220);
s3 := Sym(240)!(  1,121)(  2,122)(  3,123)(  4,124)(  5,137)(  6,138)(  7,139)
(  8,140)(  9,133)( 10,134)( 11,135)( 12,136)( 13,129)( 14,130)( 15,131)
( 16,132)( 17,125)( 18,126)( 19,127)( 20,128)( 21,141)( 22,142)( 23,143)
( 24,144)( 25,157)( 26,158)( 27,159)( 28,160)( 29,153)( 30,154)( 31,155)
( 32,156)( 33,149)( 34,150)( 35,151)( 36,152)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,161)( 42,162)( 43,163)( 44,164)( 45,177)( 46,178)( 47,179)
( 48,180)( 49,173)( 50,174)( 51,175)( 52,176)( 53,169)( 54,170)( 55,171)
( 56,172)( 57,165)( 58,166)( 59,167)( 60,168)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,197)( 66,198)( 67,199)( 68,200)( 69,193)( 70,194)( 71,195)
( 72,196)( 73,189)( 74,190)( 75,191)( 76,192)( 77,185)( 78,186)( 79,187)
( 80,188)( 81,201)( 82,202)( 83,203)( 84,204)( 85,217)( 86,218)( 87,219)
( 88,220)( 89,213)( 90,214)( 91,215)( 92,216)( 93,209)( 94,210)( 95,211)
( 96,212)( 97,205)( 98,206)( 99,207)(100,208)(101,221)(102,222)(103,223)
(104,224)(105,237)(106,238)(107,239)(108,240)(109,233)(110,234)(111,235)
(112,236)(113,229)(114,230)(115,231)(116,232)(117,225)(118,226)(119,227)
(120,228);
s4 := Sym(240)!(  1,  5)(  2,  6)(  3,  7)(  4,  8)(  9, 17)( 10, 18)( 11, 19)
( 12, 20)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 29, 37)( 30, 38)( 31, 39)
( 32, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)( 49, 57)( 50, 58)( 51, 59)
( 52, 60)( 61, 65)( 62, 66)( 63, 67)( 64, 68)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 97)( 90, 98)( 91, 99)
( 92,100)(101,105)(102,106)(103,107)(104,108)(109,117)(110,118)(111,119)
(112,120)(121,125)(122,126)(123,127)(124,128)(129,137)(130,138)(131,139)
(132,140)(141,145)(142,146)(143,147)(144,148)(149,157)(150,158)(151,159)
(152,160)(161,165)(162,166)(163,167)(164,168)(169,177)(170,178)(171,179)
(172,180)(181,185)(182,186)(183,187)(184,188)(189,197)(190,198)(191,199)
(192,200)(201,205)(202,206)(203,207)(204,208)(209,217)(210,218)(211,219)
(212,220)(221,225)(222,226)(223,227)(224,228)(229,237)(230,238)(231,239)
(232,240);
poly := sub<Sym(240)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 
References : None.
to this polytope