Polytope of Type {90,6}

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

References : None.
to this polytope