Polytope of Type {2,4,6,15}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,6,15}*1920
if this polytope has a name.
Group : SmallGroup(1920,240232)
Rank : 5
Schlafli Type : {2,4,6,15}
Number of vertices, edges, etc : 2, 4, 16, 60, 20
Order of s0s1s2s3s4 : 20
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 : {2,2,6,15}*960
5-fold quotients : {2,4,6,3}*384
10-fold quotients : {2,2,6,3}*192
12-fold quotients : {2,4,2,5}*160
20-fold quotients : {2,2,3,3}*96
24-fold quotients : {2,2,2,5}*80
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)( 9,129)( 10,130)
( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)( 17,137)( 18,138)
( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)( 25,145)( 26,146)
( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)( 33,153)( 34,154)
( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)( 41,161)( 42,162)
( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)( 49,169)( 50,170)
( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)( 57,177)( 58,178)
( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)( 65,185)( 66,186)
( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)( 73,193)( 74,194)
( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)( 81,201)( 82,202)
( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)( 88,208)( 89,209)( 90,210)
( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)( 96,216)( 97,217)( 98,218)
( 99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)(106,226)
(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)(114,234)
(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)(121,241)(122,242);;
s2 := ( 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)(123,183)(124,185)(125,184)(126,186)(127,187)(128,189)
(129,188)(130,190)(131,191)(132,193)(133,192)(134,194)(135,195)(136,197)
(137,196)(138,198)(139,199)(140,201)(141,200)(142,202)(143,223)(144,225)
(145,224)(146,226)(147,227)(148,229)(149,228)(150,230)(151,231)(152,233)
(153,232)(154,234)(155,235)(156,237)(157,236)(158,238)(159,239)(160,241)
(161,240)(162,242)(163,203)(164,205)(165,204)(166,206)(167,207)(168,209)
(169,208)(170,210)(171,211)(172,213)(173,212)(174,214)(175,215)(176,217)
(177,216)(178,218)(179,219)(180,221)(181,220)(182,222);;
s3 := ( 3, 23)( 4, 24)( 5, 26)( 6, 25)( 7, 39)( 8, 40)( 9, 42)( 10, 41)
( 11, 35)( 12, 36)( 13, 38)( 14, 37)( 15, 31)( 16, 32)( 17, 34)( 18, 33)
( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 45, 46)( 47, 59)( 48, 60)( 49, 62)
( 50, 61)( 51, 55)( 52, 56)( 53, 58)( 54, 57)( 63, 83)( 64, 84)( 65, 86)
( 66, 85)( 67, 99)( 68,100)( 69,102)( 70,101)( 71, 95)( 72, 96)( 73, 98)
( 74, 97)( 75, 91)( 76, 92)( 77, 94)( 78, 93)( 79, 87)( 80, 88)( 81, 90)
( 82, 89)(105,106)(107,119)(108,120)(109,122)(110,121)(111,115)(112,116)
(113,118)(114,117)(123,143)(124,144)(125,146)(126,145)(127,159)(128,160)
(129,162)(130,161)(131,155)(132,156)(133,158)(134,157)(135,151)(136,152)
(137,154)(138,153)(139,147)(140,148)(141,150)(142,149)(165,166)(167,179)
(168,180)(169,182)(170,181)(171,175)(172,176)(173,178)(174,177)(183,203)
(184,204)(185,206)(186,205)(187,219)(188,220)(189,222)(190,221)(191,215)
(192,216)(193,218)(194,217)(195,211)(196,212)(197,214)(198,213)(199,207)
(200,208)(201,210)(202,209)(225,226)(227,239)(228,240)(229,242)(230,241)
(231,235)(232,236)(233,238)(234,237);;
s4 := ( 3, 10)( 4, 8)( 5, 9)( 6, 7)( 11, 22)( 12, 20)( 13, 21)( 14, 19)
( 15, 18)( 23, 50)( 24, 48)( 25, 49)( 26, 47)( 27, 46)( 28, 44)( 29, 45)
( 30, 43)( 31, 62)( 32, 60)( 33, 61)( 34, 59)( 35, 58)( 36, 56)( 37, 57)
( 38, 55)( 39, 54)( 40, 52)( 41, 53)( 42, 51)( 63, 70)( 64, 68)( 65, 69)
( 66, 67)( 71, 82)( 72, 80)( 73, 81)( 74, 79)( 75, 78)( 83,110)( 84,108)
( 85,109)( 86,107)( 87,106)( 88,104)( 89,105)( 90,103)( 91,122)( 92,120)
( 93,121)( 94,119)( 95,118)( 96,116)( 97,117)( 98,115)( 99,114)(100,112)
(101,113)(102,111)(123,130)(124,128)(125,129)(126,127)(131,142)(132,140)
(133,141)(134,139)(135,138)(143,170)(144,168)(145,169)(146,167)(147,166)
(148,164)(149,165)(150,163)(151,182)(152,180)(153,181)(154,179)(155,178)
(156,176)(157,177)(158,175)(159,174)(160,172)(161,173)(162,171)(183,190)
(184,188)(185,189)(186,187)(191,202)(192,200)(193,201)(194,199)(195,198)
(203,230)(204,228)(205,229)(206,227)(207,226)(208,224)(209,225)(210,223)
(211,242)(212,240)(213,241)(214,239)(215,238)(216,236)(217,237)(218,235)
(219,234)(220,232)(221,233)(222,231);;
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*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(242)!(1,2);
s1 := Sym(242)!( 3,123)( 4,124)( 5,125)( 6,126)( 7,127)( 8,128)( 9,129)
( 10,130)( 11,131)( 12,132)( 13,133)( 14,134)( 15,135)( 16,136)( 17,137)
( 18,138)( 19,139)( 20,140)( 21,141)( 22,142)( 23,143)( 24,144)( 25,145)
( 26,146)( 27,147)( 28,148)( 29,149)( 30,150)( 31,151)( 32,152)( 33,153)
( 34,154)( 35,155)( 36,156)( 37,157)( 38,158)( 39,159)( 40,160)( 41,161)
( 42,162)( 43,163)( 44,164)( 45,165)( 46,166)( 47,167)( 48,168)( 49,169)
( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)( 57,177)
( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)( 65,185)
( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)( 73,193)
( 74,194)( 75,195)( 76,196)( 77,197)( 78,198)( 79,199)( 80,200)( 81,201)
( 82,202)( 83,203)( 84,204)( 85,205)( 86,206)( 87,207)( 88,208)( 89,209)
( 90,210)( 91,211)( 92,212)( 93,213)( 94,214)( 95,215)( 96,216)( 97,217)
( 98,218)( 99,219)(100,220)(101,221)(102,222)(103,223)(104,224)(105,225)
(106,226)(107,227)(108,228)(109,229)(110,230)(111,231)(112,232)(113,233)
(114,234)(115,235)(116,236)(117,237)(118,238)(119,239)(120,240)(121,241)
(122,242);
s2 := 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)(123,183)(124,185)(125,184)(126,186)(127,187)
(128,189)(129,188)(130,190)(131,191)(132,193)(133,192)(134,194)(135,195)
(136,197)(137,196)(138,198)(139,199)(140,201)(141,200)(142,202)(143,223)
(144,225)(145,224)(146,226)(147,227)(148,229)(149,228)(150,230)(151,231)
(152,233)(153,232)(154,234)(155,235)(156,237)(157,236)(158,238)(159,239)
(160,241)(161,240)(162,242)(163,203)(164,205)(165,204)(166,206)(167,207)
(168,209)(169,208)(170,210)(171,211)(172,213)(173,212)(174,214)(175,215)
(176,217)(177,216)(178,218)(179,219)(180,221)(181,220)(182,222);
s3 := Sym(242)!( 3, 23)( 4, 24)( 5, 26)( 6, 25)( 7, 39)( 8, 40)( 9, 42)
( 10, 41)( 11, 35)( 12, 36)( 13, 38)( 14, 37)( 15, 31)( 16, 32)( 17, 34)
( 18, 33)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 45, 46)( 47, 59)( 48, 60)
( 49, 62)( 50, 61)( 51, 55)( 52, 56)( 53, 58)( 54, 57)( 63, 83)( 64, 84)
( 65, 86)( 66, 85)( 67, 99)( 68,100)( 69,102)( 70,101)( 71, 95)( 72, 96)
( 73, 98)( 74, 97)( 75, 91)( 76, 92)( 77, 94)( 78, 93)( 79, 87)( 80, 88)
( 81, 90)( 82, 89)(105,106)(107,119)(108,120)(109,122)(110,121)(111,115)
(112,116)(113,118)(114,117)(123,143)(124,144)(125,146)(126,145)(127,159)
(128,160)(129,162)(130,161)(131,155)(132,156)(133,158)(134,157)(135,151)
(136,152)(137,154)(138,153)(139,147)(140,148)(141,150)(142,149)(165,166)
(167,179)(168,180)(169,182)(170,181)(171,175)(172,176)(173,178)(174,177)
(183,203)(184,204)(185,206)(186,205)(187,219)(188,220)(189,222)(190,221)
(191,215)(192,216)(193,218)(194,217)(195,211)(196,212)(197,214)(198,213)
(199,207)(200,208)(201,210)(202,209)(225,226)(227,239)(228,240)(229,242)
(230,241)(231,235)(232,236)(233,238)(234,237);
s4 := Sym(242)!( 3, 10)( 4, 8)( 5, 9)( 6, 7)( 11, 22)( 12, 20)( 13, 21)
( 14, 19)( 15, 18)( 23, 50)( 24, 48)( 25, 49)( 26, 47)( 27, 46)( 28, 44)
( 29, 45)( 30, 43)( 31, 62)( 32, 60)( 33, 61)( 34, 59)( 35, 58)( 36, 56)
( 37, 57)( 38, 55)( 39, 54)( 40, 52)( 41, 53)( 42, 51)( 63, 70)( 64, 68)
( 65, 69)( 66, 67)( 71, 82)( 72, 80)( 73, 81)( 74, 79)( 75, 78)( 83,110)
( 84,108)( 85,109)( 86,107)( 87,106)( 88,104)( 89,105)( 90,103)( 91,122)
( 92,120)( 93,121)( 94,119)( 95,118)( 96,116)( 97,117)( 98,115)( 99,114)
(100,112)(101,113)(102,111)(123,130)(124,128)(125,129)(126,127)(131,142)
(132,140)(133,141)(134,139)(135,138)(143,170)(144,168)(145,169)(146,167)
(147,166)(148,164)(149,165)(150,163)(151,182)(152,180)(153,181)(154,179)
(155,178)(156,176)(157,177)(158,175)(159,174)(160,172)(161,173)(162,171)
(183,190)(184,188)(185,189)(186,187)(191,202)(192,200)(193,201)(194,199)
(195,198)(203,230)(204,228)(205,229)(206,227)(207,226)(208,224)(209,225)
(210,223)(211,242)(212,240)(213,241)(214,239)(215,238)(216,236)(217,237)
(218,235)(219,234)(220,232)(221,233)(222,231);
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*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3,
s2*s3*s4*s3*s2*s3*s2*s3*s4*s3*s4*s2*s3*s4*s3*s4 >;
to this polytope