Polytope of Type {3,6,45}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,45}*1620
if this polytope has a name.
Group : SmallGroup(1620,131)
Rank : 4
Schlafli Type : {3,6,45}
Number of vertices, edges, etc : 3, 9, 135, 45
Order of s0s1s2s3 : 45
Order of s0s1s2s3s2s1 : 6
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) :
3-fold quotients : {3,2,45}*540, {3,6,15}*540
5-fold quotients : {3,6,9}*324
9-fold quotients : {3,2,15}*180
15-fold quotients : {3,2,9}*108, {3,6,3}*108
27-fold quotients : {3,2,5}*60
45-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 46, 91)
( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)
( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)( 62,108)
( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)( 70,115)
( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)( 78,122)
( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)( 86,132)
( 87,131)( 88,133)( 89,135)( 90,134)(137,138)(140,141)(143,144)(146,147)
(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)(170,171)
(173,174)(176,177)(179,180)(181,226)(182,228)(183,227)(184,229)(185,231)
(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)(193,238)
(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)(201,245)
(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)(209,255)
(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)(217,262)
(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)(225,269)
(272,273)(275,276)(278,279)(281,282)(284,285)(287,288)(290,291)(293,294)
(296,297)(299,300)(302,303)(305,306)(308,309)(311,312)(314,315)(316,361)
(317,363)(318,362)(319,364)(320,366)(321,365)(322,367)(323,369)(324,368)
(325,370)(326,372)(327,371)(328,373)(329,375)(330,374)(331,376)(332,378)
(333,377)(334,379)(335,381)(336,380)(337,382)(338,384)(339,383)(340,385)
(341,387)(342,386)(343,388)(344,390)(345,389)(346,391)(347,393)(348,392)
(349,394)(350,396)(351,395)(352,397)(353,399)(354,398)(355,400)(356,402)
(357,401)(358,403)(359,405)(360,404);;
s1 := ( 1, 46)( 2, 48)( 3, 47)( 4, 49)( 5, 51)( 6, 50)( 7, 52)( 8, 54)
( 9, 53)( 10, 55)( 11, 57)( 12, 56)( 13, 58)( 14, 60)( 15, 59)( 16, 61)
( 17, 63)( 18, 62)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)
( 25, 70)( 26, 72)( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 76)( 32, 78)
( 33, 77)( 34, 79)( 35, 81)( 36, 80)( 37, 82)( 38, 84)( 39, 83)( 40, 85)
( 41, 87)( 42, 86)( 43, 88)( 44, 90)( 45, 89)( 92, 93)( 95, 96)( 98, 99)
(101,102)(104,105)(107,108)(110,111)(113,114)(116,117)(119,120)(122,123)
(125,126)(128,129)(131,132)(134,135)(136,181)(137,183)(138,182)(139,184)
(140,186)(141,185)(142,187)(143,189)(144,188)(145,190)(146,192)(147,191)
(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,199)(155,201)
(156,200)(157,202)(158,204)(159,203)(160,205)(161,207)(162,206)(163,208)
(164,210)(165,209)(166,211)(167,213)(168,212)(169,214)(170,216)(171,215)
(172,217)(173,219)(174,218)(175,220)(176,222)(177,221)(178,223)(179,225)
(180,224)(227,228)(230,231)(233,234)(236,237)(239,240)(242,243)(245,246)
(248,249)(251,252)(254,255)(257,258)(260,261)(263,264)(266,267)(269,270)
(271,316)(272,318)(273,317)(274,319)(275,321)(276,320)(277,322)(278,324)
(279,323)(280,325)(281,327)(282,326)(283,328)(284,330)(285,329)(286,331)
(287,333)(288,332)(289,334)(290,336)(291,335)(292,337)(293,339)(294,338)
(295,340)(296,342)(297,341)(298,343)(299,345)(300,344)(301,346)(302,348)
(303,347)(304,349)(305,351)(306,350)(307,352)(308,354)(309,353)(310,355)
(311,357)(312,356)(313,358)(314,360)(315,359)(362,363)(365,366)(368,369)
(371,372)(374,375)(377,378)(380,381)(383,384)(386,387)(389,390)(392,393)
(395,396)(398,399)(401,402)(404,405);;
s2 := ( 1,136)( 2,138)( 3,137)( 4,142)( 5,144)( 6,143)( 7,139)( 8,141)
( 9,140)( 10,172)( 11,174)( 12,173)( 13,178)( 14,180)( 15,179)( 16,175)
( 17,177)( 18,176)( 19,163)( 20,165)( 21,164)( 22,169)( 23,171)( 24,170)
( 25,166)( 26,168)( 27,167)( 28,154)( 29,156)( 30,155)( 31,160)( 32,162)
( 33,161)( 34,157)( 35,159)( 36,158)( 37,145)( 38,147)( 39,146)( 40,151)
( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,182)( 47,181)( 48,183)
( 49,188)( 50,187)( 51,189)( 52,185)( 53,184)( 54,186)( 55,218)( 56,217)
( 57,219)( 58,224)( 59,223)( 60,225)( 61,221)( 62,220)( 63,222)( 64,209)
( 65,208)( 66,210)( 67,215)( 68,214)( 69,216)( 70,212)( 71,211)( 72,213)
( 73,200)( 74,199)( 75,201)( 76,206)( 77,205)( 78,207)( 79,203)( 80,202)
( 81,204)( 82,191)( 83,190)( 84,192)( 85,197)( 86,196)( 87,198)( 88,194)
( 89,193)( 90,195)( 91,228)( 92,227)( 93,226)( 94,234)( 95,233)( 96,232)
( 97,231)( 98,230)( 99,229)(100,264)(101,263)(102,262)(103,270)(104,269)
(105,268)(106,267)(107,266)(108,265)(109,255)(110,254)(111,253)(112,261)
(113,260)(114,259)(115,258)(116,257)(117,256)(118,246)(119,245)(120,244)
(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,237)(128,236)
(129,235)(130,243)(131,242)(132,241)(133,240)(134,239)(135,238)(271,274)
(272,276)(273,275)(278,279)(280,310)(281,312)(282,311)(283,307)(284,309)
(285,308)(286,313)(287,315)(288,314)(289,301)(290,303)(291,302)(292,298)
(293,300)(294,299)(295,304)(296,306)(297,305)(316,320)(317,319)(318,321)
(322,323)(325,356)(326,355)(327,357)(328,353)(329,352)(330,354)(331,359)
(332,358)(333,360)(334,347)(335,346)(336,348)(337,344)(338,343)(339,345)
(340,350)(341,349)(342,351)(361,366)(362,365)(363,364)(367,369)(370,402)
(371,401)(372,400)(373,399)(374,398)(375,397)(376,405)(377,404)(378,403)
(379,393)(380,392)(381,391)(382,390)(383,389)(384,388)(385,396)(386,395)
(387,394);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 37)( 20, 39)( 21, 38)( 22, 43)( 23, 45)( 24, 44)( 25, 40)
( 26, 42)( 27, 41)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 46, 55)( 47, 57)
( 48, 56)( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)( 64, 82)
( 65, 84)( 66, 83)( 67, 88)( 68, 90)( 69, 89)( 70, 85)( 71, 87)( 72, 86)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 91,100)( 92,102)( 93,101)( 94,106)
( 95,108)( 96,107)( 97,103)( 98,105)( 99,104)(109,127)(110,129)(111,128)
(112,133)(113,135)(114,134)(115,130)(116,132)(117,131)(119,120)(121,124)
(122,126)(123,125)(136,283)(137,285)(138,284)(139,280)(140,282)(141,281)
(142,286)(143,288)(144,287)(145,274)(146,276)(147,275)(148,271)(149,273)
(150,272)(151,277)(152,279)(153,278)(154,310)(155,312)(156,311)(157,307)
(158,309)(159,308)(160,313)(161,315)(162,314)(163,301)(164,303)(165,302)
(166,298)(167,300)(168,299)(169,304)(170,306)(171,305)(172,292)(173,294)
(174,293)(175,289)(176,291)(177,290)(178,295)(179,297)(180,296)(181,328)
(182,330)(183,329)(184,325)(185,327)(186,326)(187,331)(188,333)(189,332)
(190,319)(191,321)(192,320)(193,316)(194,318)(195,317)(196,322)(197,324)
(198,323)(199,355)(200,357)(201,356)(202,352)(203,354)(204,353)(205,358)
(206,360)(207,359)(208,346)(209,348)(210,347)(211,343)(212,345)(213,344)
(214,349)(215,351)(216,350)(217,337)(218,339)(219,338)(220,334)(221,336)
(222,335)(223,340)(224,342)(225,341)(226,373)(227,375)(228,374)(229,370)
(230,372)(231,371)(232,376)(233,378)(234,377)(235,364)(236,366)(237,365)
(238,361)(239,363)(240,362)(241,367)(242,369)(243,368)(244,400)(245,402)
(246,401)(247,397)(248,399)(249,398)(250,403)(251,405)(252,404)(253,391)
(254,393)(255,392)(256,388)(257,390)(258,389)(259,394)(260,396)(261,395)
(262,382)(263,384)(264,383)(265,379)(266,381)(267,380)(268,385)(269,387)
(270,386);;
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*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
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*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*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(405)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 46, 91)( 47, 93)( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)
( 54, 98)( 55,100)( 56,102)( 57,101)( 58,103)( 59,105)( 60,104)( 61,106)
( 62,108)( 63,107)( 64,109)( 65,111)( 66,110)( 67,112)( 68,114)( 69,113)
( 70,115)( 71,117)( 72,116)( 73,118)( 74,120)( 75,119)( 76,121)( 77,123)
( 78,122)( 79,124)( 80,126)( 81,125)( 82,127)( 83,129)( 84,128)( 85,130)
( 86,132)( 87,131)( 88,133)( 89,135)( 90,134)(137,138)(140,141)(143,144)
(146,147)(149,150)(152,153)(155,156)(158,159)(161,162)(164,165)(167,168)
(170,171)(173,174)(176,177)(179,180)(181,226)(182,228)(183,227)(184,229)
(185,231)(186,230)(187,232)(188,234)(189,233)(190,235)(191,237)(192,236)
(193,238)(194,240)(195,239)(196,241)(197,243)(198,242)(199,244)(200,246)
(201,245)(202,247)(203,249)(204,248)(205,250)(206,252)(207,251)(208,253)
(209,255)(210,254)(211,256)(212,258)(213,257)(214,259)(215,261)(216,260)
(217,262)(218,264)(219,263)(220,265)(221,267)(222,266)(223,268)(224,270)
(225,269)(272,273)(275,276)(278,279)(281,282)(284,285)(287,288)(290,291)
(293,294)(296,297)(299,300)(302,303)(305,306)(308,309)(311,312)(314,315)
(316,361)(317,363)(318,362)(319,364)(320,366)(321,365)(322,367)(323,369)
(324,368)(325,370)(326,372)(327,371)(328,373)(329,375)(330,374)(331,376)
(332,378)(333,377)(334,379)(335,381)(336,380)(337,382)(338,384)(339,383)
(340,385)(341,387)(342,386)(343,388)(344,390)(345,389)(346,391)(347,393)
(348,392)(349,394)(350,396)(351,395)(352,397)(353,399)(354,398)(355,400)
(356,402)(357,401)(358,403)(359,405)(360,404);
s1 := Sym(405)!( 1, 46)( 2, 48)( 3, 47)( 4, 49)( 5, 51)( 6, 50)( 7, 52)
( 8, 54)( 9, 53)( 10, 55)( 11, 57)( 12, 56)( 13, 58)( 14, 60)( 15, 59)
( 16, 61)( 17, 63)( 18, 62)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)
( 24, 68)( 25, 70)( 26, 72)( 27, 71)( 28, 73)( 29, 75)( 30, 74)( 31, 76)
( 32, 78)( 33, 77)( 34, 79)( 35, 81)( 36, 80)( 37, 82)( 38, 84)( 39, 83)
( 40, 85)( 41, 87)( 42, 86)( 43, 88)( 44, 90)( 45, 89)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108)(110,111)(113,114)(116,117)(119,120)
(122,123)(125,126)(128,129)(131,132)(134,135)(136,181)(137,183)(138,182)
(139,184)(140,186)(141,185)(142,187)(143,189)(144,188)(145,190)(146,192)
(147,191)(148,193)(149,195)(150,194)(151,196)(152,198)(153,197)(154,199)
(155,201)(156,200)(157,202)(158,204)(159,203)(160,205)(161,207)(162,206)
(163,208)(164,210)(165,209)(166,211)(167,213)(168,212)(169,214)(170,216)
(171,215)(172,217)(173,219)(174,218)(175,220)(176,222)(177,221)(178,223)
(179,225)(180,224)(227,228)(230,231)(233,234)(236,237)(239,240)(242,243)
(245,246)(248,249)(251,252)(254,255)(257,258)(260,261)(263,264)(266,267)
(269,270)(271,316)(272,318)(273,317)(274,319)(275,321)(276,320)(277,322)
(278,324)(279,323)(280,325)(281,327)(282,326)(283,328)(284,330)(285,329)
(286,331)(287,333)(288,332)(289,334)(290,336)(291,335)(292,337)(293,339)
(294,338)(295,340)(296,342)(297,341)(298,343)(299,345)(300,344)(301,346)
(302,348)(303,347)(304,349)(305,351)(306,350)(307,352)(308,354)(309,353)
(310,355)(311,357)(312,356)(313,358)(314,360)(315,359)(362,363)(365,366)
(368,369)(371,372)(374,375)(377,378)(380,381)(383,384)(386,387)(389,390)
(392,393)(395,396)(398,399)(401,402)(404,405);
s2 := Sym(405)!( 1,136)( 2,138)( 3,137)( 4,142)( 5,144)( 6,143)( 7,139)
( 8,141)( 9,140)( 10,172)( 11,174)( 12,173)( 13,178)( 14,180)( 15,179)
( 16,175)( 17,177)( 18,176)( 19,163)( 20,165)( 21,164)( 22,169)( 23,171)
( 24,170)( 25,166)( 26,168)( 27,167)( 28,154)( 29,156)( 30,155)( 31,160)
( 32,162)( 33,161)( 34,157)( 35,159)( 36,158)( 37,145)( 38,147)( 39,146)
( 40,151)( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,182)( 47,181)
( 48,183)( 49,188)( 50,187)( 51,189)( 52,185)( 53,184)( 54,186)( 55,218)
( 56,217)( 57,219)( 58,224)( 59,223)( 60,225)( 61,221)( 62,220)( 63,222)
( 64,209)( 65,208)( 66,210)( 67,215)( 68,214)( 69,216)( 70,212)( 71,211)
( 72,213)( 73,200)( 74,199)( 75,201)( 76,206)( 77,205)( 78,207)( 79,203)
( 80,202)( 81,204)( 82,191)( 83,190)( 84,192)( 85,197)( 86,196)( 87,198)
( 88,194)( 89,193)( 90,195)( 91,228)( 92,227)( 93,226)( 94,234)( 95,233)
( 96,232)( 97,231)( 98,230)( 99,229)(100,264)(101,263)(102,262)(103,270)
(104,269)(105,268)(106,267)(107,266)(108,265)(109,255)(110,254)(111,253)
(112,261)(113,260)(114,259)(115,258)(116,257)(117,256)(118,246)(119,245)
(120,244)(121,252)(122,251)(123,250)(124,249)(125,248)(126,247)(127,237)
(128,236)(129,235)(130,243)(131,242)(132,241)(133,240)(134,239)(135,238)
(271,274)(272,276)(273,275)(278,279)(280,310)(281,312)(282,311)(283,307)
(284,309)(285,308)(286,313)(287,315)(288,314)(289,301)(290,303)(291,302)
(292,298)(293,300)(294,299)(295,304)(296,306)(297,305)(316,320)(317,319)
(318,321)(322,323)(325,356)(326,355)(327,357)(328,353)(329,352)(330,354)
(331,359)(332,358)(333,360)(334,347)(335,346)(336,348)(337,344)(338,343)
(339,345)(340,350)(341,349)(342,351)(361,366)(362,365)(363,364)(367,369)
(370,402)(371,401)(372,400)(373,399)(374,398)(375,397)(376,405)(377,404)
(378,403)(379,393)(380,392)(381,391)(382,390)(383,389)(384,388)(385,396)
(386,395)(387,394);
s3 := Sym(405)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 37)( 20, 39)( 21, 38)( 22, 43)( 23, 45)( 24, 44)
( 25, 40)( 26, 42)( 27, 41)( 29, 30)( 31, 34)( 32, 36)( 33, 35)( 46, 55)
( 47, 57)( 48, 56)( 49, 61)( 50, 63)( 51, 62)( 52, 58)( 53, 60)( 54, 59)
( 64, 82)( 65, 84)( 66, 83)( 67, 88)( 68, 90)( 69, 89)( 70, 85)( 71, 87)
( 72, 86)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 91,100)( 92,102)( 93,101)
( 94,106)( 95,108)( 96,107)( 97,103)( 98,105)( 99,104)(109,127)(110,129)
(111,128)(112,133)(113,135)(114,134)(115,130)(116,132)(117,131)(119,120)
(121,124)(122,126)(123,125)(136,283)(137,285)(138,284)(139,280)(140,282)
(141,281)(142,286)(143,288)(144,287)(145,274)(146,276)(147,275)(148,271)
(149,273)(150,272)(151,277)(152,279)(153,278)(154,310)(155,312)(156,311)
(157,307)(158,309)(159,308)(160,313)(161,315)(162,314)(163,301)(164,303)
(165,302)(166,298)(167,300)(168,299)(169,304)(170,306)(171,305)(172,292)
(173,294)(174,293)(175,289)(176,291)(177,290)(178,295)(179,297)(180,296)
(181,328)(182,330)(183,329)(184,325)(185,327)(186,326)(187,331)(188,333)
(189,332)(190,319)(191,321)(192,320)(193,316)(194,318)(195,317)(196,322)
(197,324)(198,323)(199,355)(200,357)(201,356)(202,352)(203,354)(204,353)
(205,358)(206,360)(207,359)(208,346)(209,348)(210,347)(211,343)(212,345)
(213,344)(214,349)(215,351)(216,350)(217,337)(218,339)(219,338)(220,334)
(221,336)(222,335)(223,340)(224,342)(225,341)(226,373)(227,375)(228,374)
(229,370)(230,372)(231,371)(232,376)(233,378)(234,377)(235,364)(236,366)
(237,365)(238,361)(239,363)(240,362)(241,367)(242,369)(243,368)(244,400)
(245,402)(246,401)(247,397)(248,399)(249,398)(250,403)(251,405)(252,404)
(253,391)(254,393)(255,392)(256,388)(257,390)(258,389)(259,394)(260,396)
(261,395)(262,382)(263,384)(264,383)(265,379)(266,381)(267,380)(268,385)
(269,387)(270,386);
poly := sub<Sym(405)|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*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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*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*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