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