Polytope of Type {90,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {90,6}*1620b
if this polytope has a name.
Group : SmallGroup(1620,137)
Rank : 3
Schlafli Type : {90,6}
Number of vertices, edges, etc : 135, 405, 9
Order of s₀s₁s₂ : 45
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-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 : {30,6}*540
   5-fold quotients : {18,6}*324b
   15-fold quotients : {6,6}*108
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)(136,272)(137,271)(138,273)(139,278)(140,277)(141,279)
(142,275)(143,274)(144,276)(145,308)(146,307)(147,309)(148,314)(149,313)
(150,315)(151,311)(152,310)(153,312)(154,299)(155,298)(156,300)(157,305)
(158,304)(159,306)(160,302)(161,301)(162,303)(163,290)(164,289)(165,291)
(166,296)(167,295)(168,297)(169,293)(170,292)(171,294)(172,281)(173,280)
(174,282)(175,287)(176,286)(177,288)(178,284)(179,283)(180,285)(181,317)
(182,316)(183,318)(184,323)(185,322)(186,324)(187,320)(188,319)(189,321)
(190,353)(191,352)(192,354)(193,359)(194,358)(195,360)(196,356)(197,355)
(198,357)(199,344)(200,343)(201,345)(202,350)(203,349)(204,351)(205,347)
(206,346)(207,348)(208,335)(209,334)(210,336)(211,341)(212,340)(213,342)
(214,338)(215,337)(216,339)(217,326)(218,325)(219,327)(220,332)(221,331)
(222,333)(223,329)(224,328)(225,330)(226,362)(227,361)(228,363)(229,368)
(230,367)(231,369)(232,365)(233,364)(234,366)(235,398)(236,397)(237,399)
(238,404)(239,403)(240,405)(241,401)(242,400)(243,402)(244,389)(245,388)
(246,390)(247,395)(248,394)(249,396)(250,392)(251,391)(252,393)(253,380)
(254,379)(255,381)(256,386)(257,385)(258,387)(259,383)(260,382)(261,384)
(262,371)(263,370)(264,372)(265,377)(266,376)(267,378)(268,374)(269,373)
(270,375);;
s1 := (  1,145)(  2,147)(  3,146)(  4,148)(  5,150)(  6,149)(  7,151)(  8,153)
(  9,152)( 10,136)( 11,138)( 12,137)( 13,139)( 14,141)( 15,140)( 16,142)
( 17,144)( 18,143)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)( 24,176)
( 25,178)( 26,180)( 27,179)( 28,163)( 29,165)( 30,164)( 31,166)( 32,168)
( 33,167)( 34,169)( 35,171)( 36,170)( 37,154)( 38,156)( 39,155)( 40,157)
( 41,159)( 42,158)( 43,160)( 44,162)( 45,161)( 46,243)( 47,242)( 48,241)
( 49,237)( 50,236)( 51,235)( 52,240)( 53,239)( 54,238)( 55,234)( 56,233)
( 57,232)( 58,228)( 59,227)( 60,226)( 61,231)( 62,230)( 63,229)( 64,270)
( 65,269)( 66,268)( 67,264)( 68,263)( 69,262)( 70,267)( 71,266)( 72,265)
( 73,261)( 74,260)( 75,259)( 76,255)( 77,254)( 78,253)( 79,258)( 80,257)
( 81,256)( 82,252)( 83,251)( 84,250)( 85,246)( 86,245)( 87,244)( 88,249)
( 89,248)( 90,247)( 91,195)( 92,194)( 93,193)( 94,198)( 95,197)( 96,196)
( 97,192)( 98,191)( 99,190)(100,186)(101,185)(102,184)(103,189)(104,188)
(105,187)(106,183)(107,182)(108,181)(109,222)(110,221)(111,220)(112,225)
(113,224)(114,223)(115,219)(116,218)(117,217)(118,213)(119,212)(120,211)
(121,216)(122,215)(123,214)(124,210)(125,209)(126,208)(127,204)(128,203)
(129,202)(130,207)(131,206)(132,205)(133,201)(134,200)(135,199)(271,281)
(272,280)(273,282)(274,284)(275,283)(276,285)(277,287)(278,286)(279,288)
(289,308)(290,307)(291,309)(292,311)(293,310)(294,312)(295,314)(296,313)
(297,315)(298,299)(301,302)(304,305)(316,376)(317,378)(318,377)(319,370)
(320,372)(321,371)(322,373)(323,375)(324,374)(325,367)(326,369)(327,368)
(328,361)(329,363)(330,362)(331,364)(332,366)(333,365)(334,403)(335,405)
(336,404)(337,397)(338,399)(339,398)(340,400)(341,402)(342,401)(343,394)
(344,396)(345,395)(346,388)(347,390)(348,389)(349,391)(350,393)(351,392)
(352,385)(353,387)(354,386)(355,379)(356,381)(357,380)(358,382)(359,384)
(360,383);;
s2 := (  1, 46)(  2, 47)(  3, 48)(  4, 54)(  5, 52)(  6, 53)(  7, 50)(  8, 51)
(  9, 49)( 10, 55)( 11, 56)( 12, 57)( 13, 63)( 14, 61)( 15, 62)( 16, 59)
( 17, 60)( 18, 58)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)( 24, 71)
( 25, 68)( 26, 69)( 27, 67)( 28, 73)( 29, 74)( 30, 75)( 31, 81)( 32, 79)
( 33, 80)( 34, 77)( 35, 78)( 36, 76)( 37, 82)( 38, 83)( 39, 84)( 40, 90)
( 41, 88)( 42, 89)( 43, 86)( 44, 87)( 45, 85)( 94, 99)( 95, 97)( 96, 98)
(103,108)(104,106)(105,107)(112,117)(113,115)(114,116)(121,126)(122,124)
(123,125)(130,135)(131,133)(132,134)(136,181)(137,182)(138,183)(139,189)
(140,187)(141,188)(142,185)(143,186)(144,184)(145,190)(146,191)(147,192)
(148,198)(149,196)(150,197)(151,194)(152,195)(153,193)(154,199)(155,200)
(156,201)(157,207)(158,205)(159,206)(160,203)(161,204)(162,202)(163,208)
(164,209)(165,210)(166,216)(167,214)(168,215)(169,212)(170,213)(171,211)
(172,217)(173,218)(174,219)(175,225)(176,223)(177,224)(178,221)(179,222)
(180,220)(229,234)(230,232)(231,233)(238,243)(239,241)(240,242)(247,252)
(248,250)(249,251)(256,261)(257,259)(258,260)(265,270)(266,268)(267,269)
(271,316)(272,317)(273,318)(274,324)(275,322)(276,323)(277,320)(278,321)
(279,319)(280,325)(281,326)(282,327)(283,333)(284,331)(285,332)(286,329)
(287,330)(288,328)(289,334)(290,335)(291,336)(292,342)(293,340)(294,341)
(295,338)(296,339)(297,337)(298,343)(299,344)(300,345)(301,351)(302,349)
(303,350)(304,347)(305,348)(306,346)(307,352)(308,353)(309,354)(310,360)
(311,358)(312,359)(313,356)(314,357)(315,355)(364,369)(365,367)(366,368)
(373,378)(374,376)(375,377)(382,387)(383,385)(384,386)(391,396)(392,394)
(393,395)(400,405)(401,403)(402,404);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0 ];;
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)(136,272)(137,271)(138,273)(139,278)(140,277)
(141,279)(142,275)(143,274)(144,276)(145,308)(146,307)(147,309)(148,314)
(149,313)(150,315)(151,311)(152,310)(153,312)(154,299)(155,298)(156,300)
(157,305)(158,304)(159,306)(160,302)(161,301)(162,303)(163,290)(164,289)
(165,291)(166,296)(167,295)(168,297)(169,293)(170,292)(171,294)(172,281)
(173,280)(174,282)(175,287)(176,286)(177,288)(178,284)(179,283)(180,285)
(181,317)(182,316)(183,318)(184,323)(185,322)(186,324)(187,320)(188,319)
(189,321)(190,353)(191,352)(192,354)(193,359)(194,358)(195,360)(196,356)
(197,355)(198,357)(199,344)(200,343)(201,345)(202,350)(203,349)(204,351)
(205,347)(206,346)(207,348)(208,335)(209,334)(210,336)(211,341)(212,340)
(213,342)(214,338)(215,337)(216,339)(217,326)(218,325)(219,327)(220,332)
(221,331)(222,333)(223,329)(224,328)(225,330)(226,362)(227,361)(228,363)
(229,368)(230,367)(231,369)(232,365)(233,364)(234,366)(235,398)(236,397)
(237,399)(238,404)(239,403)(240,405)(241,401)(242,400)(243,402)(244,389)
(245,388)(246,390)(247,395)(248,394)(249,396)(250,392)(251,391)(252,393)
(253,380)(254,379)(255,381)(256,386)(257,385)(258,387)(259,383)(260,382)
(261,384)(262,371)(263,370)(264,372)(265,377)(266,376)(267,378)(268,374)
(269,373)(270,375);
s1 := Sym(405)!(  1,145)(  2,147)(  3,146)(  4,148)(  5,150)(  6,149)(  7,151)
(  8,153)(  9,152)( 10,136)( 11,138)( 12,137)( 13,139)( 14,141)( 15,140)
( 16,142)( 17,144)( 18,143)( 19,172)( 20,174)( 21,173)( 22,175)( 23,177)
( 24,176)( 25,178)( 26,180)( 27,179)( 28,163)( 29,165)( 30,164)( 31,166)
( 32,168)( 33,167)( 34,169)( 35,171)( 36,170)( 37,154)( 38,156)( 39,155)
( 40,157)( 41,159)( 42,158)( 43,160)( 44,162)( 45,161)( 46,243)( 47,242)
( 48,241)( 49,237)( 50,236)( 51,235)( 52,240)( 53,239)( 54,238)( 55,234)
( 56,233)( 57,232)( 58,228)( 59,227)( 60,226)( 61,231)( 62,230)( 63,229)
( 64,270)( 65,269)( 66,268)( 67,264)( 68,263)( 69,262)( 70,267)( 71,266)
( 72,265)( 73,261)( 74,260)( 75,259)( 76,255)( 77,254)( 78,253)( 79,258)
( 80,257)( 81,256)( 82,252)( 83,251)( 84,250)( 85,246)( 86,245)( 87,244)
( 88,249)( 89,248)( 90,247)( 91,195)( 92,194)( 93,193)( 94,198)( 95,197)
( 96,196)( 97,192)( 98,191)( 99,190)(100,186)(101,185)(102,184)(103,189)
(104,188)(105,187)(106,183)(107,182)(108,181)(109,222)(110,221)(111,220)
(112,225)(113,224)(114,223)(115,219)(116,218)(117,217)(118,213)(119,212)
(120,211)(121,216)(122,215)(123,214)(124,210)(125,209)(126,208)(127,204)
(128,203)(129,202)(130,207)(131,206)(132,205)(133,201)(134,200)(135,199)
(271,281)(272,280)(273,282)(274,284)(275,283)(276,285)(277,287)(278,286)
(279,288)(289,308)(290,307)(291,309)(292,311)(293,310)(294,312)(295,314)
(296,313)(297,315)(298,299)(301,302)(304,305)(316,376)(317,378)(318,377)
(319,370)(320,372)(321,371)(322,373)(323,375)(324,374)(325,367)(326,369)
(327,368)(328,361)(329,363)(330,362)(331,364)(332,366)(333,365)(334,403)
(335,405)(336,404)(337,397)(338,399)(339,398)(340,400)(341,402)(342,401)
(343,394)(344,396)(345,395)(346,388)(347,390)(348,389)(349,391)(350,393)
(351,392)(352,385)(353,387)(354,386)(355,379)(356,381)(357,380)(358,382)
(359,384)(360,383);
s2 := Sym(405)!(  1, 46)(  2, 47)(  3, 48)(  4, 54)(  5, 52)(  6, 53)(  7, 50)
(  8, 51)(  9, 49)( 10, 55)( 11, 56)( 12, 57)( 13, 63)( 14, 61)( 15, 62)
( 16, 59)( 17, 60)( 18, 58)( 19, 64)( 20, 65)( 21, 66)( 22, 72)( 23, 70)
( 24, 71)( 25, 68)( 26, 69)( 27, 67)( 28, 73)( 29, 74)( 30, 75)( 31, 81)
( 32, 79)( 33, 80)( 34, 77)( 35, 78)( 36, 76)( 37, 82)( 38, 83)( 39, 84)
( 40, 90)( 41, 88)( 42, 89)( 43, 86)( 44, 87)( 45, 85)( 94, 99)( 95, 97)
( 96, 98)(103,108)(104,106)(105,107)(112,117)(113,115)(114,116)(121,126)
(122,124)(123,125)(130,135)(131,133)(132,134)(136,181)(137,182)(138,183)
(139,189)(140,187)(141,188)(142,185)(143,186)(144,184)(145,190)(146,191)
(147,192)(148,198)(149,196)(150,197)(151,194)(152,195)(153,193)(154,199)
(155,200)(156,201)(157,207)(158,205)(159,206)(160,203)(161,204)(162,202)
(163,208)(164,209)(165,210)(166,216)(167,214)(168,215)(169,212)(170,213)
(171,211)(172,217)(173,218)(174,219)(175,225)(176,223)(177,224)(178,221)
(179,222)(180,220)(229,234)(230,232)(231,233)(238,243)(239,241)(240,242)
(247,252)(248,250)(249,251)(256,261)(257,259)(258,260)(265,270)(266,268)
(267,269)(271,316)(272,317)(273,318)(274,324)(275,322)(276,323)(277,320)
(278,321)(279,319)(280,325)(281,326)(282,327)(283,333)(284,331)(285,332)
(286,329)(287,330)(288,328)(289,334)(290,335)(291,336)(292,342)(293,340)
(294,341)(295,338)(296,339)(297,337)(298,343)(299,344)(300,345)(301,351)
(302,349)(303,350)(304,347)(305,348)(306,346)(307,352)(308,353)(309,354)
(310,360)(311,358)(312,359)(313,356)(314,357)(315,355)(364,369)(365,367)
(366,368)(373,378)(374,376)(375,377)(382,387)(383,385)(384,386)(391,396)
(392,394)(393,395)(400,405)(401,403)(402,404);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0 >;

References : None.
to this polytope