Polytope of Type {8,110}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,110}*1760
Also Known As : {8,110|2}. if this polytope has another name.
Group : SmallGroup(1760,1018)
Rank : 3
Schlafli Type : {8,110}
Number of vertices, edges, etc : 8, 440, 110
Order of s₀s₁s₂ : 440
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,110}*880
   4-fold quotients : {2,110}*440
   5-fold quotients : {8,22}*352
   8-fold quotients : {2,55}*220
   10-fold quotients : {4,22}*176
   11-fold quotients : {8,10}*160
   20-fold quotients : {2,22}*88
   22-fold quotients : {4,10}*80
   40-fold quotients : {2,11}*44
   44-fold quotients : {2,10}*40
   55-fold quotients : {8,2}*32
   88-fold quotients : {2,5}*20
   110-fold quotients : {4,2}*16
   220-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)
(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(125,180)(126,181)
(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)(134,189)
(135,190)(136,191)(137,192)(138,193)(139,194)(140,195)(141,196)(142,197)
(143,198)(144,199)(145,200)(146,201)(147,202)(148,203)(149,204)(150,205)
(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)(158,213)
(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)(221,331)
(222,332)(223,333)(224,334)(225,335)(226,336)(227,337)(228,338)(229,339)
(230,340)(231,341)(232,342)(233,343)(234,344)(235,345)(236,346)(237,347)
(238,348)(239,349)(240,350)(241,351)(242,352)(243,353)(244,354)(245,355)
(246,356)(247,357)(248,358)(249,359)(250,360)(251,361)(252,362)(253,363)
(254,364)(255,365)(256,366)(257,367)(258,368)(259,369)(260,370)(261,371)
(262,372)(263,373)(264,374)(265,375)(266,376)(267,377)(268,378)(269,379)
(270,380)(271,381)(272,382)(273,383)(274,384)(275,385)(276,386)(277,387)
(278,388)(279,389)(280,390)(281,391)(282,392)(283,393)(284,394)(285,395)
(286,396)(287,397)(288,398)(289,399)(290,400)(291,401)(292,402)(293,403)
(294,404)(295,405)(296,406)(297,407)(298,408)(299,409)(300,410)(301,411)
(302,412)(303,413)(304,414)(305,415)(306,416)(307,417)(308,418)(309,419)
(310,420)(311,421)(312,422)(313,423)(314,424)(315,425)(316,426)(317,427)
(318,428)(319,429)(320,430)(321,431)(322,432)(323,433)(324,434)(325,435)
(326,436)(327,437)(328,438)(329,439)(330,440);;
s1 := (  1,221)(  2,231)(  3,230)(  4,229)(  5,228)(  6,227)(  7,226)(  8,225)
(  9,224)( 10,223)( 11,222)( 12,265)( 13,275)( 14,274)( 15,273)( 16,272)
( 17,271)( 18,270)( 19,269)( 20,268)( 21,267)( 22,266)( 23,254)( 24,264)
( 25,263)( 26,262)( 27,261)( 28,260)( 29,259)( 30,258)( 31,257)( 32,256)
( 33,255)( 34,243)( 35,253)( 36,252)( 37,251)( 38,250)( 39,249)( 40,248)
( 41,247)( 42,246)( 43,245)( 44,244)( 45,232)( 46,242)( 47,241)( 48,240)
( 49,239)( 50,238)( 51,237)( 52,236)( 53,235)( 54,234)( 55,233)( 56,276)
( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)( 64,279)
( 65,278)( 66,277)( 67,320)( 68,330)( 69,329)( 70,328)( 71,327)( 72,326)
( 73,325)( 74,324)( 75,323)( 76,322)( 77,321)( 78,309)( 79,319)( 80,318)
( 81,317)( 82,316)( 83,315)( 84,314)( 85,313)( 86,312)( 87,311)( 88,310)
( 89,298)( 90,308)( 91,307)( 92,306)( 93,305)( 94,304)( 95,303)( 96,302)
( 97,301)( 98,300)( 99,299)(100,287)(101,297)(102,296)(103,295)(104,294)
(105,293)(106,292)(107,291)(108,290)(109,289)(110,288)(111,386)(112,396)
(113,395)(114,394)(115,393)(116,392)(117,391)(118,390)(119,389)(120,388)
(121,387)(122,430)(123,440)(124,439)(125,438)(126,437)(127,436)(128,435)
(129,434)(130,433)(131,432)(132,431)(133,419)(134,429)(135,428)(136,427)
(137,426)(138,425)(139,424)(140,423)(141,422)(142,421)(143,420)(144,408)
(145,418)(146,417)(147,416)(148,415)(149,414)(150,413)(151,412)(152,411)
(153,410)(154,409)(155,397)(156,407)(157,406)(158,405)(159,404)(160,403)
(161,402)(162,401)(163,400)(164,399)(165,398)(166,331)(167,341)(168,340)
(169,339)(170,338)(171,337)(172,336)(173,335)(174,334)(175,333)(176,332)
(177,375)(178,385)(179,384)(180,383)(181,382)(182,381)(183,380)(184,379)
(185,378)(186,377)(187,376)(188,364)(189,374)(190,373)(191,372)(192,371)
(193,370)(194,369)(195,368)(196,367)(197,366)(198,365)(199,353)(200,363)
(201,362)(202,361)(203,360)(204,359)(205,358)(206,357)(207,356)(208,355)
(209,354)(210,342)(211,352)(212,351)(213,350)(214,349)(215,348)(216,347)
(217,346)(218,345)(219,344)(220,343);;
s2 := (  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)
(  9, 16)( 10, 15)( 11, 14)( 23, 46)( 24, 45)( 25, 55)( 26, 54)( 27, 53)
( 28, 52)( 29, 51)( 30, 50)( 31, 49)( 32, 48)( 33, 47)( 34, 35)( 36, 44)
( 37, 43)( 38, 42)( 39, 41)( 56, 68)( 57, 67)( 58, 77)( 59, 76)( 60, 75)
( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 78,101)( 79,100)
( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)( 87,103)
( 88,102)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(111,123)(112,122)
(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)(120,125)
(121,124)(133,156)(134,155)(135,165)(136,164)(137,163)(138,162)(139,161)
(140,160)(141,159)(142,158)(143,157)(144,145)(146,154)(147,153)(148,152)
(149,151)(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)(172,183)
(173,182)(174,181)(175,180)(176,179)(188,211)(189,210)(190,220)(191,219)
(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)(199,200)
(201,209)(202,208)(203,207)(204,206)(221,233)(222,232)(223,242)(224,241)
(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(243,266)
(244,265)(245,275)(246,274)(247,273)(248,272)(249,271)(250,270)(251,269)
(252,268)(253,267)(254,255)(256,264)(257,263)(258,262)(259,261)(276,288)
(277,287)(278,297)(279,296)(280,295)(281,294)(282,293)(283,292)(284,291)
(285,290)(286,289)(298,321)(299,320)(300,330)(301,329)(302,328)(303,327)
(304,326)(305,325)(306,324)(307,323)(308,322)(309,310)(311,319)(312,318)
(313,317)(314,316)(331,343)(332,342)(333,352)(334,351)(335,350)(336,349)
(337,348)(338,347)(339,346)(340,345)(341,344)(353,376)(354,375)(355,385)
(356,384)(357,383)(358,382)(359,381)(360,380)(361,379)(362,378)(363,377)
(364,365)(366,374)(367,373)(368,372)(369,371)(386,398)(387,397)(388,407)
(389,406)(390,405)(391,404)(392,403)(393,402)(394,401)(395,400)(396,399)
(408,431)(409,430)(410,440)(411,439)(412,438)(413,437)(414,436)(415,435)
(416,434)(417,433)(418,432)(419,420)(421,429)(422,428)(423,427)(424,426);;
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*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(440)!(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)
(118,173)(119,174)(120,175)(121,176)(122,177)(123,178)(124,179)(125,180)
(126,181)(127,182)(128,183)(129,184)(130,185)(131,186)(132,187)(133,188)
(134,189)(135,190)(136,191)(137,192)(138,193)(139,194)(140,195)(141,196)
(142,197)(143,198)(144,199)(145,200)(146,201)(147,202)(148,203)(149,204)
(150,205)(151,206)(152,207)(153,208)(154,209)(155,210)(156,211)(157,212)
(158,213)(159,214)(160,215)(161,216)(162,217)(163,218)(164,219)(165,220)
(221,331)(222,332)(223,333)(224,334)(225,335)(226,336)(227,337)(228,338)
(229,339)(230,340)(231,341)(232,342)(233,343)(234,344)(235,345)(236,346)
(237,347)(238,348)(239,349)(240,350)(241,351)(242,352)(243,353)(244,354)
(245,355)(246,356)(247,357)(248,358)(249,359)(250,360)(251,361)(252,362)
(253,363)(254,364)(255,365)(256,366)(257,367)(258,368)(259,369)(260,370)
(261,371)(262,372)(263,373)(264,374)(265,375)(266,376)(267,377)(268,378)
(269,379)(270,380)(271,381)(272,382)(273,383)(274,384)(275,385)(276,386)
(277,387)(278,388)(279,389)(280,390)(281,391)(282,392)(283,393)(284,394)
(285,395)(286,396)(287,397)(288,398)(289,399)(290,400)(291,401)(292,402)
(293,403)(294,404)(295,405)(296,406)(297,407)(298,408)(299,409)(300,410)
(301,411)(302,412)(303,413)(304,414)(305,415)(306,416)(307,417)(308,418)
(309,419)(310,420)(311,421)(312,422)(313,423)(314,424)(315,425)(316,426)
(317,427)(318,428)(319,429)(320,430)(321,431)(322,432)(323,433)(324,434)
(325,435)(326,436)(327,437)(328,438)(329,439)(330,440);
s1 := Sym(440)!(  1,221)(  2,231)(  3,230)(  4,229)(  5,228)(  6,227)(  7,226)
(  8,225)(  9,224)( 10,223)( 11,222)( 12,265)( 13,275)( 14,274)( 15,273)
( 16,272)( 17,271)( 18,270)( 19,269)( 20,268)( 21,267)( 22,266)( 23,254)
( 24,264)( 25,263)( 26,262)( 27,261)( 28,260)( 29,259)( 30,258)( 31,257)
( 32,256)( 33,255)( 34,243)( 35,253)( 36,252)( 37,251)( 38,250)( 39,249)
( 40,248)( 41,247)( 42,246)( 43,245)( 44,244)( 45,232)( 46,242)( 47,241)
( 48,240)( 49,239)( 50,238)( 51,237)( 52,236)( 53,235)( 54,234)( 55,233)
( 56,276)( 57,286)( 58,285)( 59,284)( 60,283)( 61,282)( 62,281)( 63,280)
( 64,279)( 65,278)( 66,277)( 67,320)( 68,330)( 69,329)( 70,328)( 71,327)
( 72,326)( 73,325)( 74,324)( 75,323)( 76,322)( 77,321)( 78,309)( 79,319)
( 80,318)( 81,317)( 82,316)( 83,315)( 84,314)( 85,313)( 86,312)( 87,311)
( 88,310)( 89,298)( 90,308)( 91,307)( 92,306)( 93,305)( 94,304)( 95,303)
( 96,302)( 97,301)( 98,300)( 99,299)(100,287)(101,297)(102,296)(103,295)
(104,294)(105,293)(106,292)(107,291)(108,290)(109,289)(110,288)(111,386)
(112,396)(113,395)(114,394)(115,393)(116,392)(117,391)(118,390)(119,389)
(120,388)(121,387)(122,430)(123,440)(124,439)(125,438)(126,437)(127,436)
(128,435)(129,434)(130,433)(131,432)(132,431)(133,419)(134,429)(135,428)
(136,427)(137,426)(138,425)(139,424)(140,423)(141,422)(142,421)(143,420)
(144,408)(145,418)(146,417)(147,416)(148,415)(149,414)(150,413)(151,412)
(152,411)(153,410)(154,409)(155,397)(156,407)(157,406)(158,405)(159,404)
(160,403)(161,402)(162,401)(163,400)(164,399)(165,398)(166,331)(167,341)
(168,340)(169,339)(170,338)(171,337)(172,336)(173,335)(174,334)(175,333)
(176,332)(177,375)(178,385)(179,384)(180,383)(181,382)(182,381)(183,380)
(184,379)(185,378)(186,377)(187,376)(188,364)(189,374)(190,373)(191,372)
(192,371)(193,370)(194,369)(195,368)(196,367)(197,366)(198,365)(199,353)
(200,363)(201,362)(202,361)(203,360)(204,359)(205,358)(206,357)(207,356)
(208,355)(209,354)(210,342)(211,352)(212,351)(213,350)(214,349)(215,348)
(216,347)(217,346)(218,345)(219,344)(220,343);
s2 := Sym(440)!(  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)
(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 23, 46)( 24, 45)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 51)( 30, 50)( 31, 49)( 32, 48)( 33, 47)( 34, 35)
( 36, 44)( 37, 43)( 38, 42)( 39, 41)( 56, 68)( 57, 67)( 58, 77)( 59, 76)
( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 78,101)
( 79,100)( 80,110)( 81,109)( 82,108)( 83,107)( 84,106)( 85,105)( 86,104)
( 87,103)( 88,102)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)(111,123)
(112,122)(113,132)(114,131)(115,130)(116,129)(117,128)(118,127)(119,126)
(120,125)(121,124)(133,156)(134,155)(135,165)(136,164)(137,163)(138,162)
(139,161)(140,160)(141,159)(142,158)(143,157)(144,145)(146,154)(147,153)
(148,152)(149,151)(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)
(172,183)(173,182)(174,181)(175,180)(176,179)(188,211)(189,210)(190,220)
(191,219)(192,218)(193,217)(194,216)(195,215)(196,214)(197,213)(198,212)
(199,200)(201,209)(202,208)(203,207)(204,206)(221,233)(222,232)(223,242)
(224,241)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)
(243,266)(244,265)(245,275)(246,274)(247,273)(248,272)(249,271)(250,270)
(251,269)(252,268)(253,267)(254,255)(256,264)(257,263)(258,262)(259,261)
(276,288)(277,287)(278,297)(279,296)(280,295)(281,294)(282,293)(283,292)
(284,291)(285,290)(286,289)(298,321)(299,320)(300,330)(301,329)(302,328)
(303,327)(304,326)(305,325)(306,324)(307,323)(308,322)(309,310)(311,319)
(312,318)(313,317)(314,316)(331,343)(332,342)(333,352)(334,351)(335,350)
(336,349)(337,348)(338,347)(339,346)(340,345)(341,344)(353,376)(354,375)
(355,385)(356,384)(357,383)(358,382)(359,381)(360,380)(361,379)(362,378)
(363,377)(364,365)(366,374)(367,373)(368,372)(369,371)(386,398)(387,397)
(388,407)(389,406)(390,405)(391,404)(392,403)(393,402)(394,401)(395,400)
(396,399)(408,431)(409,430)(410,440)(411,439)(412,438)(413,437)(414,436)
(415,435)(416,434)(417,433)(418,432)(419,420)(421,429)(422,428)(423,427)
(424,426);
poly := sub<Sym(440)|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*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope