Polytope of Type {15,6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {15,6,8}*1440
if this polytope has a name.
Group : SmallGroup(1440,3811)
Rank : 4
Schlafli Type : {15,6,8}
Number of vertices, edges, etc : 15, 45, 24, 8
Order of s0s1s2s3 : 120
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 : {15,6,4}*720
3-fold quotients : {15,2,8}*480
4-fold quotients : {15,6,2}*360
5-fold quotients : {3,6,8}*288
6-fold quotients : {15,2,4}*240
9-fold quotients : {5,2,8}*160
10-fold quotients : {3,6,4}*144
12-fold quotients : {15,2,2}*120
15-fold quotients : {3,2,8}*96
18-fold quotients : {5,2,4}*80
20-fold quotients : {3,6,2}*72
30-fold quotients : {3,2,4}*48
36-fold quotients : {5,2,2}*40
60-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)( 16, 31)
( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 42)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 47, 50)( 48, 49)
( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 76)( 62, 80)( 63, 79)
( 64, 78)( 65, 77)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)
( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 92, 95)( 93, 94)( 96,101)( 97,105)
( 98,104)( 99,103)(100,102)(106,121)(107,125)(108,124)(109,123)(110,122)
(111,131)(112,135)(113,134)(114,133)(115,132)(116,126)(117,130)(118,129)
(119,128)(120,127)(137,140)(138,139)(141,146)(142,150)(143,149)(144,148)
(145,147)(151,166)(152,170)(153,169)(154,168)(155,167)(156,176)(157,180)
(158,179)(159,178)(160,177)(161,171)(162,175)(163,174)(164,173)(165,172)
(182,185)(183,184)(186,191)(187,195)(188,194)(189,193)(190,192)(196,211)
(197,215)(198,214)(199,213)(200,212)(201,221)(202,225)(203,224)(204,223)
(205,222)(206,216)(207,220)(208,219)(209,218)(210,217)(227,230)(228,229)
(231,236)(232,240)(233,239)(234,238)(235,237)(241,256)(242,260)(243,259)
(244,258)(245,257)(246,266)(247,270)(248,269)(249,268)(250,267)(251,261)
(252,265)(253,264)(254,263)(255,262)(272,275)(273,274)(276,281)(277,285)
(278,284)(279,283)(280,282)(286,301)(287,305)(288,304)(289,303)(290,302)
(291,311)(292,315)(293,314)(294,313)(295,312)(296,306)(297,310)(298,309)
(299,308)(300,307)(317,320)(318,319)(321,326)(322,330)(323,329)(324,328)
(325,327)(331,346)(332,350)(333,349)(334,348)(335,347)(336,356)(337,360)
(338,359)(339,358)(340,357)(341,351)(342,355)(343,354)(344,353)(345,352);;
s1 := ( 1, 22)( 2, 21)( 3, 25)( 4, 24)( 5, 23)( 6, 17)( 7, 16)( 8, 20)
( 9, 19)( 10, 18)( 11, 27)( 12, 26)( 13, 30)( 14, 29)( 15, 28)( 31, 37)
( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 67)( 47, 66)
( 48, 70)( 49, 69)( 50, 68)( 51, 62)( 52, 61)( 53, 65)( 54, 64)( 55, 63)
( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 76, 82)( 77, 81)( 78, 85)
( 79, 84)( 80, 83)( 86, 87)( 88, 90)( 91,112)( 92,111)( 93,115)( 94,114)
( 95,113)( 96,107)( 97,106)( 98,110)( 99,109)(100,108)(101,117)(102,116)
(103,120)(104,119)(105,118)(121,127)(122,126)(123,130)(124,129)(125,128)
(131,132)(133,135)(136,157)(137,156)(138,160)(139,159)(140,158)(141,152)
(142,151)(143,155)(144,154)(145,153)(146,162)(147,161)(148,165)(149,164)
(150,163)(166,172)(167,171)(168,175)(169,174)(170,173)(176,177)(178,180)
(181,202)(182,201)(183,205)(184,204)(185,203)(186,197)(187,196)(188,200)
(189,199)(190,198)(191,207)(192,206)(193,210)(194,209)(195,208)(211,217)
(212,216)(213,220)(214,219)(215,218)(221,222)(223,225)(226,247)(227,246)
(228,250)(229,249)(230,248)(231,242)(232,241)(233,245)(234,244)(235,243)
(236,252)(237,251)(238,255)(239,254)(240,253)(256,262)(257,261)(258,265)
(259,264)(260,263)(266,267)(268,270)(271,292)(272,291)(273,295)(274,294)
(275,293)(276,287)(277,286)(278,290)(279,289)(280,288)(281,297)(282,296)
(283,300)(284,299)(285,298)(301,307)(302,306)(303,310)(304,309)(305,308)
(311,312)(313,315)(316,337)(317,336)(318,340)(319,339)(320,338)(321,332)
(322,331)(323,335)(324,334)(325,333)(326,342)(327,341)(328,345)(329,344)
(330,343)(346,352)(347,351)(348,355)(349,354)(350,353)(356,357)(358,360);;
s2 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)
( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)
( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,136)( 92,137)
( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)(100,145)
(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)(108,168)
(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)(116,176)
(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)(124,154)
(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)(132,162)
(133,163)(134,164)(135,165)(181,271)(182,272)(183,273)(184,274)(185,275)
(186,276)(187,277)(188,278)(189,279)(190,280)(191,281)(192,282)(193,283)
(194,284)(195,285)(196,301)(197,302)(198,303)(199,304)(200,305)(201,306)
(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,313)(209,314)
(210,315)(211,286)(212,287)(213,288)(214,289)(215,290)(216,291)(217,292)
(218,293)(219,294)(220,295)(221,296)(222,297)(223,298)(224,299)(225,300)
(226,316)(227,317)(228,318)(229,319)(230,320)(231,321)(232,322)(233,323)
(234,324)(235,325)(236,326)(237,327)(238,328)(239,329)(240,330)(241,346)
(242,347)(243,348)(244,349)(245,350)(246,351)(247,352)(248,353)(249,354)
(250,355)(251,356)(252,357)(253,358)(254,359)(255,360)(256,331)(257,332)
(258,333)(259,334)(260,335)(261,336)(262,337)(263,338)(264,339)(265,340)
(266,341)(267,342)(268,343)(269,344)(270,345);;
s3 := ( 1,181)( 2,182)( 3,183)( 4,184)( 5,185)( 6,186)( 7,187)( 8,188)
( 9,189)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)( 16,196)
( 17,197)( 18,198)( 19,199)( 20,200)( 21,201)( 22,202)( 23,203)( 24,204)
( 25,205)( 26,206)( 27,207)( 28,208)( 29,209)( 30,210)( 31,211)( 32,212)
( 33,213)( 34,214)( 35,215)( 36,216)( 37,217)( 38,218)( 39,219)( 40,220)
( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,226)( 47,227)( 48,228)
( 49,229)( 50,230)( 51,231)( 52,232)( 53,233)( 54,234)( 55,235)( 56,236)
( 57,237)( 58,238)( 59,239)( 60,240)( 61,241)( 62,242)( 63,243)( 64,244)
( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)( 72,252)
( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)( 80,260)
( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)( 88,268)
( 89,269)( 90,270)( 91,316)( 92,317)( 93,318)( 94,319)( 95,320)( 96,321)
( 97,322)( 98,323)( 99,324)(100,325)(101,326)(102,327)(103,328)(104,329)
(105,330)(106,331)(107,332)(108,333)(109,334)(110,335)(111,336)(112,337)
(113,338)(114,339)(115,340)(116,341)(117,342)(118,343)(119,344)(120,345)
(121,346)(122,347)(123,348)(124,349)(125,350)(126,351)(127,352)(128,353)
(129,354)(130,355)(131,356)(132,357)(133,358)(134,359)(135,360)(136,271)
(137,272)(138,273)(139,274)(140,275)(141,276)(142,277)(143,278)(144,279)
(145,280)(146,281)(147,282)(148,283)(149,284)(150,285)(151,286)(152,287)
(153,288)(154,289)(155,290)(156,291)(157,292)(158,293)(159,294)(160,295)
(161,296)(162,297)(163,298)(164,299)(165,300)(166,301)(167,302)(168,303)
(169,304)(170,305)(171,306)(172,307)(173,308)(174,309)(175,310)(176,311)
(177,312)(178,313)(179,314)(180,315);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(360)!( 2, 5)( 3, 4)( 6, 11)( 7, 15)( 8, 14)( 9, 13)( 10, 12)
( 16, 31)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 41)( 22, 45)( 23, 44)
( 24, 43)( 25, 42)( 26, 36)( 27, 40)( 28, 39)( 29, 38)( 30, 37)( 47, 50)
( 48, 49)( 51, 56)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 61, 76)( 62, 80)
( 63, 79)( 64, 78)( 65, 77)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)
( 71, 81)( 72, 85)( 73, 84)( 74, 83)( 75, 82)( 92, 95)( 93, 94)( 96,101)
( 97,105)( 98,104)( 99,103)(100,102)(106,121)(107,125)(108,124)(109,123)
(110,122)(111,131)(112,135)(113,134)(114,133)(115,132)(116,126)(117,130)
(118,129)(119,128)(120,127)(137,140)(138,139)(141,146)(142,150)(143,149)
(144,148)(145,147)(151,166)(152,170)(153,169)(154,168)(155,167)(156,176)
(157,180)(158,179)(159,178)(160,177)(161,171)(162,175)(163,174)(164,173)
(165,172)(182,185)(183,184)(186,191)(187,195)(188,194)(189,193)(190,192)
(196,211)(197,215)(198,214)(199,213)(200,212)(201,221)(202,225)(203,224)
(204,223)(205,222)(206,216)(207,220)(208,219)(209,218)(210,217)(227,230)
(228,229)(231,236)(232,240)(233,239)(234,238)(235,237)(241,256)(242,260)
(243,259)(244,258)(245,257)(246,266)(247,270)(248,269)(249,268)(250,267)
(251,261)(252,265)(253,264)(254,263)(255,262)(272,275)(273,274)(276,281)
(277,285)(278,284)(279,283)(280,282)(286,301)(287,305)(288,304)(289,303)
(290,302)(291,311)(292,315)(293,314)(294,313)(295,312)(296,306)(297,310)
(298,309)(299,308)(300,307)(317,320)(318,319)(321,326)(322,330)(323,329)
(324,328)(325,327)(331,346)(332,350)(333,349)(334,348)(335,347)(336,356)
(337,360)(338,359)(339,358)(340,357)(341,351)(342,355)(343,354)(344,353)
(345,352);
s1 := Sym(360)!( 1, 22)( 2, 21)( 3, 25)( 4, 24)( 5, 23)( 6, 17)( 7, 16)
( 8, 20)( 9, 19)( 10, 18)( 11, 27)( 12, 26)( 13, 30)( 14, 29)( 15, 28)
( 31, 37)( 32, 36)( 33, 40)( 34, 39)( 35, 38)( 41, 42)( 43, 45)( 46, 67)
( 47, 66)( 48, 70)( 49, 69)( 50, 68)( 51, 62)( 52, 61)( 53, 65)( 54, 64)
( 55, 63)( 56, 72)( 57, 71)( 58, 75)( 59, 74)( 60, 73)( 76, 82)( 77, 81)
( 78, 85)( 79, 84)( 80, 83)( 86, 87)( 88, 90)( 91,112)( 92,111)( 93,115)
( 94,114)( 95,113)( 96,107)( 97,106)( 98,110)( 99,109)(100,108)(101,117)
(102,116)(103,120)(104,119)(105,118)(121,127)(122,126)(123,130)(124,129)
(125,128)(131,132)(133,135)(136,157)(137,156)(138,160)(139,159)(140,158)
(141,152)(142,151)(143,155)(144,154)(145,153)(146,162)(147,161)(148,165)
(149,164)(150,163)(166,172)(167,171)(168,175)(169,174)(170,173)(176,177)
(178,180)(181,202)(182,201)(183,205)(184,204)(185,203)(186,197)(187,196)
(188,200)(189,199)(190,198)(191,207)(192,206)(193,210)(194,209)(195,208)
(211,217)(212,216)(213,220)(214,219)(215,218)(221,222)(223,225)(226,247)
(227,246)(228,250)(229,249)(230,248)(231,242)(232,241)(233,245)(234,244)
(235,243)(236,252)(237,251)(238,255)(239,254)(240,253)(256,262)(257,261)
(258,265)(259,264)(260,263)(266,267)(268,270)(271,292)(272,291)(273,295)
(274,294)(275,293)(276,287)(277,286)(278,290)(279,289)(280,288)(281,297)
(282,296)(283,300)(284,299)(285,298)(301,307)(302,306)(303,310)(304,309)
(305,308)(311,312)(313,315)(316,337)(317,336)(318,340)(319,339)(320,338)
(321,332)(322,331)(323,335)(324,334)(325,333)(326,342)(327,341)(328,345)
(329,344)(330,343)(346,352)(347,351)(348,355)(349,354)(350,353)(356,357)
(358,360);
s2 := Sym(360)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)( 91,136)
( 92,137)( 93,138)( 94,139)( 95,140)( 96,141)( 97,142)( 98,143)( 99,144)
(100,145)(101,146)(102,147)(103,148)(104,149)(105,150)(106,166)(107,167)
(108,168)(109,169)(110,170)(111,171)(112,172)(113,173)(114,174)(115,175)
(116,176)(117,177)(118,178)(119,179)(120,180)(121,151)(122,152)(123,153)
(124,154)(125,155)(126,156)(127,157)(128,158)(129,159)(130,160)(131,161)
(132,162)(133,163)(134,164)(135,165)(181,271)(182,272)(183,273)(184,274)
(185,275)(186,276)(187,277)(188,278)(189,279)(190,280)(191,281)(192,282)
(193,283)(194,284)(195,285)(196,301)(197,302)(198,303)(199,304)(200,305)
(201,306)(202,307)(203,308)(204,309)(205,310)(206,311)(207,312)(208,313)
(209,314)(210,315)(211,286)(212,287)(213,288)(214,289)(215,290)(216,291)
(217,292)(218,293)(219,294)(220,295)(221,296)(222,297)(223,298)(224,299)
(225,300)(226,316)(227,317)(228,318)(229,319)(230,320)(231,321)(232,322)
(233,323)(234,324)(235,325)(236,326)(237,327)(238,328)(239,329)(240,330)
(241,346)(242,347)(243,348)(244,349)(245,350)(246,351)(247,352)(248,353)
(249,354)(250,355)(251,356)(252,357)(253,358)(254,359)(255,360)(256,331)
(257,332)(258,333)(259,334)(260,335)(261,336)(262,337)(263,338)(264,339)
(265,340)(266,341)(267,342)(268,343)(269,344)(270,345);
s3 := Sym(360)!( 1,181)( 2,182)( 3,183)( 4,184)( 5,185)( 6,186)( 7,187)
( 8,188)( 9,189)( 10,190)( 11,191)( 12,192)( 13,193)( 14,194)( 15,195)
( 16,196)( 17,197)( 18,198)( 19,199)( 20,200)( 21,201)( 22,202)( 23,203)
( 24,204)( 25,205)( 26,206)( 27,207)( 28,208)( 29,209)( 30,210)( 31,211)
( 32,212)( 33,213)( 34,214)( 35,215)( 36,216)( 37,217)( 38,218)( 39,219)
( 40,220)( 41,221)( 42,222)( 43,223)( 44,224)( 45,225)( 46,226)( 47,227)
( 48,228)( 49,229)( 50,230)( 51,231)( 52,232)( 53,233)( 54,234)( 55,235)
( 56,236)( 57,237)( 58,238)( 59,239)( 60,240)( 61,241)( 62,242)( 63,243)
( 64,244)( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)
( 72,252)( 73,253)( 74,254)( 75,255)( 76,256)( 77,257)( 78,258)( 79,259)
( 80,260)( 81,261)( 82,262)( 83,263)( 84,264)( 85,265)( 86,266)( 87,267)
( 88,268)( 89,269)( 90,270)( 91,316)( 92,317)( 93,318)( 94,319)( 95,320)
( 96,321)( 97,322)( 98,323)( 99,324)(100,325)(101,326)(102,327)(103,328)
(104,329)(105,330)(106,331)(107,332)(108,333)(109,334)(110,335)(111,336)
(112,337)(113,338)(114,339)(115,340)(116,341)(117,342)(118,343)(119,344)
(120,345)(121,346)(122,347)(123,348)(124,349)(125,350)(126,351)(127,352)
(128,353)(129,354)(130,355)(131,356)(132,357)(133,358)(134,359)(135,360)
(136,271)(137,272)(138,273)(139,274)(140,275)(141,276)(142,277)(143,278)
(144,279)(145,280)(146,281)(147,282)(148,283)(149,284)(150,285)(151,286)
(152,287)(153,288)(154,289)(155,290)(156,291)(157,292)(158,293)(159,294)
(160,295)(161,296)(162,297)(163,298)(164,299)(165,300)(166,301)(167,302)
(168,303)(169,304)(170,305)(171,306)(172,307)(173,308)(174,309)(175,310)
(176,311)(177,312)(178,313)(179,314)(180,315);
poly := sub<Sym(360)|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,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope