Polytope of Type {90,6}

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

References : None.
to this polytope