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