Polytope of Type {45,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,6}*1620a
if this polytope has a name.
Group : SmallGroup(1620,131)
Rank : 3
Schlafli Type : {45,6}
Number of vertices, edges, etc : 135, 405, 18
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {45,6}*540, {15,6}*540
   5-fold quotients : {9,6}*324a
   9-fold quotients : {45,2}*180, {15,6}*180
   15-fold quotients : {9,6}*108, {3,6}*108
   27-fold quotients : {15,2}*60
   45-fold quotients : {9,2}*36, {3,6}*36
   81-fold quotients : {5,2}*20
   135-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 37)( 11, 38)( 12, 39)( 13, 43)( 14, 44)
( 15, 45)( 16, 40)( 17, 41)( 18, 42)( 19, 28)( 20, 29)( 21, 30)( 22, 34)
( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 46, 91)( 47, 92)( 48, 93)
( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)( 55,127)( 56,128)
( 57,129)( 58,133)( 59,134)( 60,135)( 61,130)( 62,131)( 63,132)( 64,118)
( 65,119)( 66,120)( 67,124)( 68,125)( 69,126)( 70,121)( 71,122)( 72,123)
( 73,109)( 74,110)( 75,111)( 76,115)( 77,116)( 78,117)( 79,112)( 80,113)
( 81,114)( 82,100)( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)
( 89,104)( 90,105)(136,274)(137,275)(138,276)(139,271)(140,272)(141,273)
(142,277)(143,278)(144,279)(145,310)(146,311)(147,312)(148,307)(149,308)
(150,309)(151,313)(152,314)(153,315)(154,301)(155,302)(156,303)(157,298)
(158,299)(159,300)(160,304)(161,305)(162,306)(163,292)(164,293)(165,294)
(166,289)(167,290)(168,291)(169,295)(170,296)(171,297)(172,283)(173,284)
(174,285)(175,280)(176,281)(177,282)(178,286)(179,287)(180,288)(181,364)
(182,365)(183,366)(184,361)(185,362)(186,363)(187,367)(188,368)(189,369)
(190,400)(191,401)(192,402)(193,397)(194,398)(195,399)(196,403)(197,404)
(198,405)(199,391)(200,392)(201,393)(202,388)(203,389)(204,390)(205,394)
(206,395)(207,396)(208,382)(209,383)(210,384)(211,379)(212,380)(213,381)
(214,385)(215,386)(216,387)(217,373)(218,374)(219,375)(220,370)(221,371)
(222,372)(223,376)(224,377)(225,378)(226,319)(227,320)(228,321)(229,316)
(230,317)(231,318)(232,322)(233,323)(234,324)(235,355)(236,356)(237,357)
(238,352)(239,353)(240,354)(241,358)(242,359)(243,360)(244,346)(245,347)
(246,348)(247,343)(248,344)(249,345)(250,349)(251,350)(252,351)(253,337)
(254,338)(255,339)(256,334)(257,335)(258,336)(259,340)(260,341)(261,342)
(262,328)(263,329)(264,330)(265,325)(266,326)(267,327)(268,331)(269,332)
(270,333);;
s1 := (  1,326)(  2,327)(  3,325)(  4,332)(  5,333)(  6,331)(  7,329)(  8,330)
(  9,328)( 10,317)( 11,318)( 12,316)( 13,323)( 14,324)( 15,322)( 16,320)
( 17,321)( 18,319)( 19,353)( 20,354)( 21,352)( 22,359)( 23,360)( 24,358)
( 25,356)( 26,357)( 27,355)( 28,344)( 29,345)( 30,343)( 31,350)( 32,351)
( 33,349)( 34,347)( 35,348)( 36,346)( 37,335)( 38,336)( 39,334)( 40,341)
( 41,342)( 42,340)( 43,338)( 44,339)( 45,337)( 46,282)( 47,280)( 48,281)
( 49,288)( 50,286)( 51,287)( 52,285)( 53,283)( 54,284)( 55,273)( 56,271)
( 57,272)( 58,279)( 59,277)( 60,278)( 61,276)( 62,274)( 63,275)( 64,309)
( 65,307)( 66,308)( 67,315)( 68,313)( 69,314)( 70,312)( 71,310)( 72,311)
( 73,300)( 74,298)( 75,299)( 76,306)( 77,304)( 78,305)( 79,303)( 80,301)
( 81,302)( 82,291)( 83,289)( 84,290)( 85,297)( 86,295)( 87,296)( 88,294)
( 89,292)( 90,293)( 91,370)( 92,371)( 93,372)( 94,376)( 95,377)( 96,378)
( 97,373)( 98,374)( 99,375)(100,361)(101,362)(102,363)(103,367)(104,368)
(105,369)(106,364)(107,365)(108,366)(109,397)(110,398)(111,399)(112,403)
(113,404)(114,405)(115,400)(116,401)(117,402)(118,388)(119,389)(120,390)
(121,394)(122,395)(123,396)(124,391)(125,392)(126,393)(127,379)(128,380)
(129,381)(130,385)(131,386)(132,387)(133,382)(134,383)(135,384)(136,191)
(137,192)(138,190)(139,197)(140,198)(141,196)(142,194)(143,195)(144,193)
(145,182)(146,183)(147,181)(148,188)(149,189)(150,187)(151,185)(152,186)
(153,184)(154,218)(155,219)(156,217)(157,224)(158,225)(159,223)(160,221)
(161,222)(162,220)(163,209)(164,210)(165,208)(166,215)(167,216)(168,214)
(169,212)(170,213)(171,211)(172,200)(173,201)(174,199)(175,206)(176,207)
(177,205)(178,203)(179,204)(180,202)(226,235)(227,236)(228,237)(229,241)
(230,242)(231,243)(232,238)(233,239)(234,240)(244,262)(245,263)(246,264)
(247,268)(248,269)(249,270)(250,265)(251,266)(252,267)(256,259)(257,260)
(258,261);;
s2 := (  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]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*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*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(405)!(  4,  7)(  5,  8)(  6,  9)( 10, 37)( 11, 38)( 12, 39)( 13, 43)
( 14, 44)( 15, 45)( 16, 40)( 17, 41)( 18, 42)( 19, 28)( 20, 29)( 21, 30)
( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 46, 91)( 47, 92)
( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)( 55,127)
( 56,128)( 57,129)( 58,133)( 59,134)( 60,135)( 61,130)( 62,131)( 63,132)
( 64,118)( 65,119)( 66,120)( 67,124)( 68,125)( 69,126)( 70,121)( 71,122)
( 72,123)( 73,109)( 74,110)( 75,111)( 76,115)( 77,116)( 78,117)( 79,112)
( 80,113)( 81,114)( 82,100)( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)
( 88,103)( 89,104)( 90,105)(136,274)(137,275)(138,276)(139,271)(140,272)
(141,273)(142,277)(143,278)(144,279)(145,310)(146,311)(147,312)(148,307)
(149,308)(150,309)(151,313)(152,314)(153,315)(154,301)(155,302)(156,303)
(157,298)(158,299)(159,300)(160,304)(161,305)(162,306)(163,292)(164,293)
(165,294)(166,289)(167,290)(168,291)(169,295)(170,296)(171,297)(172,283)
(173,284)(174,285)(175,280)(176,281)(177,282)(178,286)(179,287)(180,288)
(181,364)(182,365)(183,366)(184,361)(185,362)(186,363)(187,367)(188,368)
(189,369)(190,400)(191,401)(192,402)(193,397)(194,398)(195,399)(196,403)
(197,404)(198,405)(199,391)(200,392)(201,393)(202,388)(203,389)(204,390)
(205,394)(206,395)(207,396)(208,382)(209,383)(210,384)(211,379)(212,380)
(213,381)(214,385)(215,386)(216,387)(217,373)(218,374)(219,375)(220,370)
(221,371)(222,372)(223,376)(224,377)(225,378)(226,319)(227,320)(228,321)
(229,316)(230,317)(231,318)(232,322)(233,323)(234,324)(235,355)(236,356)
(237,357)(238,352)(239,353)(240,354)(241,358)(242,359)(243,360)(244,346)
(245,347)(246,348)(247,343)(248,344)(249,345)(250,349)(251,350)(252,351)
(253,337)(254,338)(255,339)(256,334)(257,335)(258,336)(259,340)(260,341)
(261,342)(262,328)(263,329)(264,330)(265,325)(266,326)(267,327)(268,331)
(269,332)(270,333);
s1 := Sym(405)!(  1,326)(  2,327)(  3,325)(  4,332)(  5,333)(  6,331)(  7,329)
(  8,330)(  9,328)( 10,317)( 11,318)( 12,316)( 13,323)( 14,324)( 15,322)
( 16,320)( 17,321)( 18,319)( 19,353)( 20,354)( 21,352)( 22,359)( 23,360)
( 24,358)( 25,356)( 26,357)( 27,355)( 28,344)( 29,345)( 30,343)( 31,350)
( 32,351)( 33,349)( 34,347)( 35,348)( 36,346)( 37,335)( 38,336)( 39,334)
( 40,341)( 41,342)( 42,340)( 43,338)( 44,339)( 45,337)( 46,282)( 47,280)
( 48,281)( 49,288)( 50,286)( 51,287)( 52,285)( 53,283)( 54,284)( 55,273)
( 56,271)( 57,272)( 58,279)( 59,277)( 60,278)( 61,276)( 62,274)( 63,275)
( 64,309)( 65,307)( 66,308)( 67,315)( 68,313)( 69,314)( 70,312)( 71,310)
( 72,311)( 73,300)( 74,298)( 75,299)( 76,306)( 77,304)( 78,305)( 79,303)
( 80,301)( 81,302)( 82,291)( 83,289)( 84,290)( 85,297)( 86,295)( 87,296)
( 88,294)( 89,292)( 90,293)( 91,370)( 92,371)( 93,372)( 94,376)( 95,377)
( 96,378)( 97,373)( 98,374)( 99,375)(100,361)(101,362)(102,363)(103,367)
(104,368)(105,369)(106,364)(107,365)(108,366)(109,397)(110,398)(111,399)
(112,403)(113,404)(114,405)(115,400)(116,401)(117,402)(118,388)(119,389)
(120,390)(121,394)(122,395)(123,396)(124,391)(125,392)(126,393)(127,379)
(128,380)(129,381)(130,385)(131,386)(132,387)(133,382)(134,383)(135,384)
(136,191)(137,192)(138,190)(139,197)(140,198)(141,196)(142,194)(143,195)
(144,193)(145,182)(146,183)(147,181)(148,188)(149,189)(150,187)(151,185)
(152,186)(153,184)(154,218)(155,219)(156,217)(157,224)(158,225)(159,223)
(160,221)(161,222)(162,220)(163,209)(164,210)(165,208)(166,215)(167,216)
(168,214)(169,212)(170,213)(171,211)(172,200)(173,201)(174,199)(175,206)
(176,207)(177,205)(178,203)(179,204)(180,202)(226,235)(227,236)(228,237)
(229,241)(230,242)(231,243)(232,238)(233,239)(234,240)(244,262)(245,263)
(246,264)(247,268)(248,269)(249,270)(250,265)(251,266)(252,267)(256,259)
(257,260)(258,261);
s2 := 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>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*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*s0*s1 >;

References : None.
to this polytope