Polytope of Type {45,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {45,6}*1620d
if this polytope has a name.
Group : SmallGroup(1620,137)
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 : {15,6}*540
   5-fold quotients : {9,6}*324d
   9-fold quotients : {15,6}*180
   15-fold quotients : {3,6}*108
   27-fold quotients : {15,2}*60
   45-fold quotients : {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 := (  2,  3)(  5,  6)(  8,  9)( 10, 37)( 11, 39)( 12, 38)( 13, 40)( 14, 42)
( 15, 41)( 16, 43)( 17, 45)( 18, 44)( 19, 28)( 20, 30)( 21, 29)( 22, 31)
( 23, 33)( 24, 32)( 25, 34)( 26, 36)( 27, 35)( 46, 91)( 47, 93)( 48, 92)
( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98)( 55,127)( 56,129)
( 57,128)( 58,130)( 59,132)( 60,131)( 61,133)( 62,135)( 63,134)( 64,118)
( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)( 72,125)
( 73,109)( 74,111)( 75,110)( 76,112)( 77,114)( 78,113)( 79,115)( 80,117)
( 81,116)( 82,100)( 83,102)( 84,101)( 85,103)( 86,105)( 87,104)( 88,106)
( 89,108)( 90,107)(136,272)(137,271)(138,273)(139,275)(140,274)(141,276)
(142,278)(143,277)(144,279)(145,308)(146,307)(147,309)(148,311)(149,310)
(150,312)(151,314)(152,313)(153,315)(154,299)(155,298)(156,300)(157,302)
(158,301)(159,303)(160,305)(161,304)(162,306)(163,290)(164,289)(165,291)
(166,293)(167,292)(168,294)(169,296)(170,295)(171,297)(172,281)(173,280)
(174,282)(175,284)(176,283)(177,285)(178,287)(179,286)(180,288)(181,362)
(182,361)(183,363)(184,365)(185,364)(186,366)(187,368)(188,367)(189,369)
(190,398)(191,397)(192,399)(193,401)(194,400)(195,402)(196,404)(197,403)
(198,405)(199,389)(200,388)(201,390)(202,392)(203,391)(204,393)(205,395)
(206,394)(207,396)(208,380)(209,379)(210,381)(211,383)(212,382)(213,384)
(214,386)(215,385)(216,387)(217,371)(218,370)(219,372)(220,374)(221,373)
(222,375)(223,377)(224,376)(225,378)(226,317)(227,316)(228,318)(229,320)
(230,319)(231,321)(232,323)(233,322)(234,324)(235,353)(236,352)(237,354)
(238,356)(239,355)(240,357)(241,359)(242,358)(243,360)(244,344)(245,343)
(246,345)(247,347)(248,346)(249,348)(250,350)(251,349)(252,351)(253,335)
(254,334)(255,336)(256,338)(257,337)(258,339)(259,341)(260,340)(261,342)
(262,326)(263,325)(264,327)(265,329)(266,328)(267,330)(268,332)(269,331)
(270,333);;
s1 := (  1,328)(  2,330)(  3,329)(  4,332)(  5,331)(  6,333)(  7,327)(  8,326)
(  9,325)( 10,319)( 11,321)( 12,320)( 13,323)( 14,322)( 15,324)( 16,318)
( 17,317)( 18,316)( 19,355)( 20,357)( 21,356)( 22,359)( 23,358)( 24,360)
( 25,354)( 26,353)( 27,352)( 28,346)( 29,348)( 30,347)( 31,350)( 32,349)
( 33,351)( 34,345)( 35,344)( 36,343)( 37,337)( 38,339)( 39,338)( 40,341)
( 41,340)( 42,342)( 43,336)( 44,335)( 45,334)( 46,288)( 47,287)( 48,286)
( 49,280)( 50,282)( 51,281)( 52,284)( 53,283)( 54,285)( 55,279)( 56,278)
( 57,277)( 58,271)( 59,273)( 60,272)( 61,275)( 62,274)( 63,276)( 64,315)
( 65,314)( 66,313)( 67,307)( 68,309)( 69,308)( 70,311)( 71,310)( 72,312)
( 73,306)( 74,305)( 75,304)( 76,298)( 77,300)( 78,299)( 79,302)( 80,301)
( 81,303)( 82,297)( 83,296)( 84,295)( 85,289)( 86,291)( 87,290)( 88,293)
( 89,292)( 90,294)( 91,370)( 92,372)( 93,371)( 94,374)( 95,373)( 96,375)
( 97,378)( 98,377)( 99,376)(100,361)(101,363)(102,362)(103,365)(104,364)
(105,366)(106,369)(107,368)(108,367)(109,397)(110,399)(111,398)(112,401)
(113,400)(114,402)(115,405)(116,404)(117,403)(118,388)(119,390)(120,389)
(121,392)(122,391)(123,393)(124,396)(125,395)(126,394)(127,379)(128,381)
(129,380)(130,383)(131,382)(132,384)(133,387)(134,386)(135,385)(136,193)
(137,195)(138,194)(139,197)(140,196)(141,198)(142,192)(143,191)(144,190)
(145,184)(146,186)(147,185)(148,188)(149,187)(150,189)(151,183)(152,182)
(153,181)(154,220)(155,222)(156,221)(157,224)(158,223)(159,225)(160,219)
(161,218)(162,217)(163,211)(164,213)(165,212)(166,215)(167,214)(168,216)
(169,210)(170,209)(171,208)(172,202)(173,204)(174,203)(175,206)(176,205)
(177,207)(178,201)(179,200)(180,199)(226,235)(227,237)(228,236)(229,239)
(230,238)(231,240)(232,243)(233,242)(234,241)(244,262)(245,264)(246,263)
(247,266)(248,265)(249,267)(250,270)(251,269)(252,268)(254,255)(256,257)
(259,261);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 46, 91)
( 47, 92)( 48, 93)( 49, 97)( 50, 98)( 51, 99)( 52, 94)( 53, 95)( 54, 96)
( 55,100)( 56,101)( 57,102)( 58,106)( 59,107)( 60,108)( 61,103)( 62,104)
( 63,105)( 64,109)( 65,110)( 66,111)( 67,115)( 68,116)( 69,117)( 70,112)
( 71,113)( 72,114)( 73,118)( 74,119)( 75,120)( 76,124)( 77,125)( 78,126)
( 79,121)( 80,122)( 81,123)( 82,127)( 83,128)( 84,129)( 85,133)( 86,134)
( 87,135)( 88,130)( 89,131)( 90,132)(139,142)(140,143)(141,144)(148,151)
(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)(168,171)
(175,178)(176,179)(177,180)(181,226)(182,227)(183,228)(184,232)(185,233)
(186,234)(187,229)(188,230)(189,231)(190,235)(191,236)(192,237)(193,241)
(194,242)(195,243)(196,238)(197,239)(198,240)(199,244)(200,245)(201,246)
(202,250)(203,251)(204,252)(205,247)(206,248)(207,249)(208,253)(209,254)
(210,255)(211,259)(212,260)(213,261)(214,256)(215,257)(216,258)(217,262)
(218,263)(219,264)(220,268)(221,269)(222,270)(223,265)(224,266)(225,267)
(274,277)(275,278)(276,279)(283,286)(284,287)(285,288)(292,295)(293,296)
(294,297)(301,304)(302,305)(303,306)(310,313)(311,314)(312,315)(316,361)
(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)(323,365)(324,366)
(325,370)(326,371)(327,372)(328,376)(329,377)(330,378)(331,373)(332,374)
(333,375)(334,379)(335,380)(336,381)(337,385)(338,386)(339,387)(340,382)
(341,383)(342,384)(343,388)(344,389)(345,390)(346,394)(347,395)(348,396)
(349,391)(350,392)(351,393)(352,397)(353,398)(354,399)(355,403)(356,404)
(357,405)(358,400)(359,401)(360,402);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope