Polytope of Type {2,3,4,20}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,4,20}*1920
if this polytope has a name.
Group : SmallGroup(1920,240142)
Rank : 5
Schlafli Type : {2,3,4,20}
Number of vertices, edges, etc : 2, 6, 12, 80, 20
Order of s0s1s2s3s4 : 60
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,3,4,10}*960
4-fold quotients : {2,3,2,20}*480
5-fold quotients : {2,3,4,4}*384b
8-fold quotients : {2,3,2,10}*240
10-fold quotients : {2,3,4,2}*192
16-fold quotients : {2,3,2,5}*120
20-fold quotients : {2,3,2,4}*96, {2,3,4,2}*96
40-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 23, 43)( 24, 45)( 25, 44)
( 26, 46)( 27, 47)( 28, 49)( 29, 48)( 30, 50)( 31, 51)( 32, 53)( 33, 52)
( 34, 54)( 35, 55)( 36, 57)( 37, 56)( 38, 58)( 39, 59)( 40, 61)( 41, 60)
( 42, 62)( 64, 65)( 68, 69)( 72, 73)( 76, 77)( 80, 81)( 83,103)( 84,105)
( 85,104)( 86,106)( 87,107)( 88,109)( 89,108)( 90,110)( 91,111)( 92,113)
( 93,112)( 94,114)( 95,115)( 96,117)( 97,116)( 98,118)( 99,119)(100,121)
(101,120)(102,122)(124,125)(128,129)(132,133)(136,137)(140,141)(143,163)
(144,165)(145,164)(146,166)(147,167)(148,169)(149,168)(150,170)(151,171)
(152,173)(153,172)(154,174)(155,175)(156,177)(157,176)(158,178)(159,179)
(160,181)(161,180)(162,182)(184,185)(188,189)(192,193)(196,197)(200,201)
(203,223)(204,225)(205,224)(206,226)(207,227)(208,229)(209,228)(210,230)
(211,231)(212,233)(213,232)(214,234)(215,235)(216,237)(217,236)(218,238)
(219,239)(220,241)(221,240)(222,242);;
s2 := ( 3, 23)( 4, 24)( 5, 26)( 6, 25)( 7, 27)( 8, 28)( 9, 30)( 10, 29)
( 11, 31)( 12, 32)( 13, 34)( 14, 33)( 15, 35)( 16, 36)( 17, 38)( 18, 37)
( 19, 39)( 20, 40)( 21, 42)( 22, 41)( 45, 46)( 49, 50)( 53, 54)( 57, 58)
( 61, 62)( 63, 83)( 64, 84)( 65, 86)( 66, 85)( 67, 87)( 68, 88)( 69, 90)
( 70, 89)( 71, 91)( 72, 92)( 73, 94)( 74, 93)( 75, 95)( 76, 96)( 77, 98)
( 78, 97)( 79, 99)( 80,100)( 81,102)( 82,101)(105,106)(109,110)(113,114)
(117,118)(121,122)(123,143)(124,144)(125,146)(126,145)(127,147)(128,148)
(129,150)(130,149)(131,151)(132,152)(133,154)(134,153)(135,155)(136,156)
(137,158)(138,157)(139,159)(140,160)(141,162)(142,161)(165,166)(169,170)
(173,174)(177,178)(181,182)(183,203)(184,204)(185,206)(186,205)(187,207)
(188,208)(189,210)(190,209)(191,211)(192,212)(193,214)(194,213)(195,215)
(196,216)(197,218)(198,217)(199,219)(200,220)(201,222)(202,221)(225,226)
(229,230)(233,234)(237,238)(241,242);;
s3 := ( 3, 6)( 4, 5)( 7, 22)( 8, 21)( 9, 20)( 10, 19)( 11, 18)( 12, 17)
( 13, 16)( 14, 15)( 23, 26)( 24, 25)( 27, 42)( 28, 41)( 29, 40)( 30, 39)
( 31, 38)( 32, 37)( 33, 36)( 34, 35)( 43, 46)( 44, 45)( 47, 62)( 48, 61)
( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)( 63, 66)( 64, 65)
( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 78)( 72, 77)( 73, 76)( 74, 75)
( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)
( 93, 96)( 94, 95)(103,106)(104,105)(107,122)(108,121)(109,120)(110,119)
(111,118)(112,117)(113,116)(114,115)(123,186)(124,185)(125,184)(126,183)
(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)(134,195)
(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)(142,187)
(143,206)(144,205)(145,204)(146,203)(147,222)(148,221)(149,220)(150,219)
(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)(158,211)
(159,210)(160,209)(161,208)(162,207)(163,226)(164,225)(165,224)(166,223)
(167,242)(168,241)(169,240)(170,239)(171,238)(172,237)(173,236)(174,235)
(175,234)(176,233)(177,232)(178,231)(179,230)(180,229)(181,228)(182,227);;
s4 := ( 3,127)( 4,128)( 5,129)( 6,130)( 7,123)( 8,124)( 9,125)( 10,126)
( 11,139)( 12,140)( 13,141)( 14,142)( 15,135)( 16,136)( 17,137)( 18,138)
( 19,131)( 20,132)( 21,133)( 22,134)( 23,147)( 24,148)( 25,149)( 26,150)
( 27,143)( 28,144)( 29,145)( 30,146)( 31,159)( 32,160)( 33,161)( 34,162)
( 35,155)( 36,156)( 37,157)( 38,158)( 39,151)( 40,152)( 41,153)( 42,154)
( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)( 50,166)
( 51,179)( 52,180)( 53,181)( 54,182)( 55,175)( 56,176)( 57,177)( 58,178)
( 59,171)( 60,172)( 61,173)( 62,174)( 63,187)( 64,188)( 65,189)( 66,190)
( 67,183)( 68,184)( 69,185)( 70,186)( 71,199)( 72,200)( 73,201)( 74,202)
( 75,195)( 76,196)( 77,197)( 78,198)( 79,191)( 80,192)( 81,193)( 82,194)
( 83,207)( 84,208)( 85,209)( 86,210)( 87,203)( 88,204)( 89,205)( 90,206)
( 91,219)( 92,220)( 93,221)( 94,222)( 95,215)( 96,216)( 97,217)( 98,218)
( 99,211)(100,212)(101,213)(102,214)(103,227)(104,228)(105,229)(106,230)
(107,223)(108,224)(109,225)(110,226)(111,239)(112,240)(113,241)(114,242)
(115,235)(116,236)(117,237)(118,238)(119,231)(120,232)(121,233)(122,234);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*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(242)!(1,2);
s1 := Sym(242)!( 4, 5)( 8, 9)( 12, 13)( 16, 17)( 20, 21)( 23, 43)( 24, 45)
( 25, 44)( 26, 46)( 27, 47)( 28, 49)( 29, 48)( 30, 50)( 31, 51)( 32, 53)
( 33, 52)( 34, 54)( 35, 55)( 36, 57)( 37, 56)( 38, 58)( 39, 59)( 40, 61)
( 41, 60)( 42, 62)( 64, 65)( 68, 69)( 72, 73)( 76, 77)( 80, 81)( 83,103)
( 84,105)( 85,104)( 86,106)( 87,107)( 88,109)( 89,108)( 90,110)( 91,111)
( 92,113)( 93,112)( 94,114)( 95,115)( 96,117)( 97,116)( 98,118)( 99,119)
(100,121)(101,120)(102,122)(124,125)(128,129)(132,133)(136,137)(140,141)
(143,163)(144,165)(145,164)(146,166)(147,167)(148,169)(149,168)(150,170)
(151,171)(152,173)(153,172)(154,174)(155,175)(156,177)(157,176)(158,178)
(159,179)(160,181)(161,180)(162,182)(184,185)(188,189)(192,193)(196,197)
(200,201)(203,223)(204,225)(205,224)(206,226)(207,227)(208,229)(209,228)
(210,230)(211,231)(212,233)(213,232)(214,234)(215,235)(216,237)(217,236)
(218,238)(219,239)(220,241)(221,240)(222,242);
s2 := Sym(242)!( 3, 23)( 4, 24)( 5, 26)( 6, 25)( 7, 27)( 8, 28)( 9, 30)
( 10, 29)( 11, 31)( 12, 32)( 13, 34)( 14, 33)( 15, 35)( 16, 36)( 17, 38)
( 18, 37)( 19, 39)( 20, 40)( 21, 42)( 22, 41)( 45, 46)( 49, 50)( 53, 54)
( 57, 58)( 61, 62)( 63, 83)( 64, 84)( 65, 86)( 66, 85)( 67, 87)( 68, 88)
( 69, 90)( 70, 89)( 71, 91)( 72, 92)( 73, 94)( 74, 93)( 75, 95)( 76, 96)
( 77, 98)( 78, 97)( 79, 99)( 80,100)( 81,102)( 82,101)(105,106)(109,110)
(113,114)(117,118)(121,122)(123,143)(124,144)(125,146)(126,145)(127,147)
(128,148)(129,150)(130,149)(131,151)(132,152)(133,154)(134,153)(135,155)
(136,156)(137,158)(138,157)(139,159)(140,160)(141,162)(142,161)(165,166)
(169,170)(173,174)(177,178)(181,182)(183,203)(184,204)(185,206)(186,205)
(187,207)(188,208)(189,210)(190,209)(191,211)(192,212)(193,214)(194,213)
(195,215)(196,216)(197,218)(198,217)(199,219)(200,220)(201,222)(202,221)
(225,226)(229,230)(233,234)(237,238)(241,242);
s3 := Sym(242)!( 3, 6)( 4, 5)( 7, 22)( 8, 21)( 9, 20)( 10, 19)( 11, 18)
( 12, 17)( 13, 16)( 14, 15)( 23, 26)( 24, 25)( 27, 42)( 28, 41)( 29, 40)
( 30, 39)( 31, 38)( 32, 37)( 33, 36)( 34, 35)( 43, 46)( 44, 45)( 47, 62)
( 48, 61)( 49, 60)( 50, 59)( 51, 58)( 52, 57)( 53, 56)( 54, 55)( 63, 66)
( 64, 65)( 67, 82)( 68, 81)( 69, 80)( 70, 79)( 71, 78)( 72, 77)( 73, 76)
( 74, 75)( 83, 86)( 84, 85)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)
( 92, 97)( 93, 96)( 94, 95)(103,106)(104,105)(107,122)(108,121)(109,120)
(110,119)(111,118)(112,117)(113,116)(114,115)(123,186)(124,185)(125,184)
(126,183)(127,202)(128,201)(129,200)(130,199)(131,198)(132,197)(133,196)
(134,195)(135,194)(136,193)(137,192)(138,191)(139,190)(140,189)(141,188)
(142,187)(143,206)(144,205)(145,204)(146,203)(147,222)(148,221)(149,220)
(150,219)(151,218)(152,217)(153,216)(154,215)(155,214)(156,213)(157,212)
(158,211)(159,210)(160,209)(161,208)(162,207)(163,226)(164,225)(165,224)
(166,223)(167,242)(168,241)(169,240)(170,239)(171,238)(172,237)(173,236)
(174,235)(175,234)(176,233)(177,232)(178,231)(179,230)(180,229)(181,228)
(182,227);
s4 := Sym(242)!( 3,127)( 4,128)( 5,129)( 6,130)( 7,123)( 8,124)( 9,125)
( 10,126)( 11,139)( 12,140)( 13,141)( 14,142)( 15,135)( 16,136)( 17,137)
( 18,138)( 19,131)( 20,132)( 21,133)( 22,134)( 23,147)( 24,148)( 25,149)
( 26,150)( 27,143)( 28,144)( 29,145)( 30,146)( 31,159)( 32,160)( 33,161)
( 34,162)( 35,155)( 36,156)( 37,157)( 38,158)( 39,151)( 40,152)( 41,153)
( 42,154)( 43,167)( 44,168)( 45,169)( 46,170)( 47,163)( 48,164)( 49,165)
( 50,166)( 51,179)( 52,180)( 53,181)( 54,182)( 55,175)( 56,176)( 57,177)
( 58,178)( 59,171)( 60,172)( 61,173)( 62,174)( 63,187)( 64,188)( 65,189)
( 66,190)( 67,183)( 68,184)( 69,185)( 70,186)( 71,199)( 72,200)( 73,201)
( 74,202)( 75,195)( 76,196)( 77,197)( 78,198)( 79,191)( 80,192)( 81,193)
( 82,194)( 83,207)( 84,208)( 85,209)( 86,210)( 87,203)( 88,204)( 89,205)
( 90,206)( 91,219)( 92,220)( 93,221)( 94,222)( 95,215)( 96,216)( 97,217)
( 98,218)( 99,211)(100,212)(101,213)(102,214)(103,227)(104,228)(105,229)
(106,230)(107,223)(108,224)(109,225)(110,226)(111,239)(112,240)(113,241)
(114,242)(115,235)(116,236)(117,237)(118,238)(119,231)(120,232)(121,233)
(122,234);
poly := sub<Sym(242)|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*s1*s0*s1, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
to this polytope