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