Polytope of Type {4,117}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,117}*936
if this polytope has a name.
Group : SmallGroup(936,58)
Rank : 3
Schlafli Type : {4,117}
Number of vertices, edges, etc : 4, 234, 117
Order of s0s1s2 : 117
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,117,2} of size 1872
Vertex Figure Of :
{2,4,117} of size 1872
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,39}*312
13-fold quotients : {4,9}*72
39-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,117}*1872, {4,234}*1872b, {4,234}*1872c
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)(222,224)
(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)(238,240)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)(254,256)
(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)(270,272)
(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)(286,288)
(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)(318,320)
(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)(334,336)
(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)(350,352)
(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)(366,368)
(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)(382,384)
(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)(398,400)
(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)(414,416)
(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)(430,432)
(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)(446,448)
(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)(462,464)
(465,467)(466,468);;
s1 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13,145)( 14,147)( 15,146)
( 16,148)( 17,153)( 18,155)( 19,154)( 20,156)( 21,149)( 22,151)( 23,150)
( 24,152)( 25,133)( 26,135)( 27,134)( 28,136)( 29,141)( 30,143)( 31,142)
( 32,144)( 33,137)( 34,139)( 35,138)( 36,140)( 37,121)( 38,123)( 39,122)
( 40,124)( 41,129)( 42,131)( 43,130)( 44,132)( 45,125)( 46,127)( 47,126)
( 48,128)( 49,109)( 50,111)( 51,110)( 52,112)( 53,117)( 54,119)( 55,118)
( 56,120)( 57,113)( 58,115)( 59,114)( 60,116)( 61, 97)( 62, 99)( 63, 98)
( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)( 71,102)
( 72,104)( 73, 85)( 74, 87)( 75, 86)( 76, 88)( 77, 93)( 78, 95)( 79, 94)
( 80, 96)( 81, 89)( 82, 91)( 83, 90)( 84, 92)(157,317)(158,319)(159,318)
(160,320)(161,313)(162,315)(163,314)(164,316)(165,321)(166,323)(167,322)
(168,324)(169,461)(170,463)(171,462)(172,464)(173,457)(174,459)(175,458)
(176,460)(177,465)(178,467)(179,466)(180,468)(181,449)(182,451)(183,450)
(184,452)(185,445)(186,447)(187,446)(188,448)(189,453)(190,455)(191,454)
(192,456)(193,437)(194,439)(195,438)(196,440)(197,433)(198,435)(199,434)
(200,436)(201,441)(202,443)(203,442)(204,444)(205,425)(206,427)(207,426)
(208,428)(209,421)(210,423)(211,422)(212,424)(213,429)(214,431)(215,430)
(216,432)(217,413)(218,415)(219,414)(220,416)(221,409)(222,411)(223,410)
(224,412)(225,417)(226,419)(227,418)(228,420)(229,401)(230,403)(231,402)
(232,404)(233,397)(234,399)(235,398)(236,400)(237,405)(238,407)(239,406)
(240,408)(241,389)(242,391)(243,390)(244,392)(245,385)(246,387)(247,386)
(248,388)(249,393)(250,395)(251,394)(252,396)(253,377)(254,379)(255,378)
(256,380)(257,373)(258,375)(259,374)(260,376)(261,381)(262,383)(263,382)
(264,384)(265,365)(266,367)(267,366)(268,368)(269,361)(270,363)(271,362)
(272,364)(273,369)(274,371)(275,370)(276,372)(277,353)(278,355)(279,354)
(280,356)(281,349)(282,351)(283,350)(284,352)(285,357)(286,359)(287,358)
(288,360)(289,341)(290,343)(291,342)(292,344)(293,337)(294,339)(295,338)
(296,340)(297,345)(298,347)(299,346)(300,348)(301,329)(302,331)(303,330)
(304,332)(305,325)(306,327)(307,326)(308,328)(309,333)(310,335)(311,334)
(312,336);;
s2 := ( 1,325)( 2,328)( 3,327)( 4,326)( 5,333)( 6,336)( 7,335)( 8,334)
( 9,329)( 10,332)( 11,331)( 12,330)( 13,313)( 14,316)( 15,315)( 16,314)
( 17,321)( 18,324)( 19,323)( 20,322)( 21,317)( 22,320)( 23,319)( 24,318)
( 25,457)( 26,460)( 27,459)( 28,458)( 29,465)( 30,468)( 31,467)( 32,466)
( 33,461)( 34,464)( 35,463)( 36,462)( 37,445)( 38,448)( 39,447)( 40,446)
( 41,453)( 42,456)( 43,455)( 44,454)( 45,449)( 46,452)( 47,451)( 48,450)
( 49,433)( 50,436)( 51,435)( 52,434)( 53,441)( 54,444)( 55,443)( 56,442)
( 57,437)( 58,440)( 59,439)( 60,438)( 61,421)( 62,424)( 63,423)( 64,422)
( 65,429)( 66,432)( 67,431)( 68,430)( 69,425)( 70,428)( 71,427)( 72,426)
( 73,409)( 74,412)( 75,411)( 76,410)( 77,417)( 78,420)( 79,419)( 80,418)
( 81,413)( 82,416)( 83,415)( 84,414)( 85,397)( 86,400)( 87,399)( 88,398)
( 89,405)( 90,408)( 91,407)( 92,406)( 93,401)( 94,404)( 95,403)( 96,402)
( 97,385)( 98,388)( 99,387)(100,386)(101,393)(102,396)(103,395)(104,394)
(105,389)(106,392)(107,391)(108,390)(109,373)(110,376)(111,375)(112,374)
(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)(120,378)
(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)(128,370)
(129,365)(130,368)(131,367)(132,366)(133,349)(134,352)(135,351)(136,350)
(137,357)(138,360)(139,359)(140,358)(141,353)(142,356)(143,355)(144,354)
(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)(152,346)
(153,341)(154,344)(155,343)(156,342)(157,169)(158,172)(159,171)(160,170)
(161,177)(162,180)(163,179)(164,178)(165,173)(166,176)(167,175)(168,174)
(181,301)(182,304)(183,303)(184,302)(185,309)(186,312)(187,311)(188,310)
(189,305)(190,308)(191,307)(192,306)(193,289)(194,292)(195,291)(196,290)
(197,297)(198,300)(199,299)(200,298)(201,293)(202,296)(203,295)(204,294)
(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)(211,287)(212,286)
(213,281)(214,284)(215,283)(216,282)(217,265)(218,268)(219,267)(220,266)
(221,273)(222,276)(223,275)(224,274)(225,269)(226,272)(227,271)(228,270)
(229,253)(230,256)(231,255)(232,254)(233,261)(234,264)(235,263)(236,262)
(237,257)(238,260)(239,259)(240,258)(242,244)(245,249)(246,252)(247,251)
(248,250);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(468)!( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,219)(218,220)(221,223)
(222,224)(225,227)(226,228)(229,231)(230,232)(233,235)(234,236)(237,239)
(238,240)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,255)
(254,256)(257,259)(258,260)(261,263)(262,264)(265,267)(266,268)(269,271)
(270,272)(273,275)(274,276)(277,279)(278,280)(281,283)(282,284)(285,287)
(286,288)(289,291)(290,292)(293,295)(294,296)(297,299)(298,300)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,315)(314,316)(317,319)
(318,320)(321,323)(322,324)(325,327)(326,328)(329,331)(330,332)(333,335)
(334,336)(337,339)(338,340)(341,343)(342,344)(345,347)(346,348)(349,351)
(350,352)(353,355)(354,356)(357,359)(358,360)(361,363)(362,364)(365,367)
(366,368)(369,371)(370,372)(373,375)(374,376)(377,379)(378,380)(381,383)
(382,384)(385,387)(386,388)(389,391)(390,392)(393,395)(394,396)(397,399)
(398,400)(401,403)(402,404)(405,407)(406,408)(409,411)(410,412)(413,415)
(414,416)(417,419)(418,420)(421,423)(422,424)(425,427)(426,428)(429,431)
(430,432)(433,435)(434,436)(437,439)(438,440)(441,443)(442,444)(445,447)
(446,448)(449,451)(450,452)(453,455)(454,456)(457,459)(458,460)(461,463)
(462,464)(465,467)(466,468);
s1 := Sym(468)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13,145)( 14,147)
( 15,146)( 16,148)( 17,153)( 18,155)( 19,154)( 20,156)( 21,149)( 22,151)
( 23,150)( 24,152)( 25,133)( 26,135)( 27,134)( 28,136)( 29,141)( 30,143)
( 31,142)( 32,144)( 33,137)( 34,139)( 35,138)( 36,140)( 37,121)( 38,123)
( 39,122)( 40,124)( 41,129)( 42,131)( 43,130)( 44,132)( 45,125)( 46,127)
( 47,126)( 48,128)( 49,109)( 50,111)( 51,110)( 52,112)( 53,117)( 54,119)
( 55,118)( 56,120)( 57,113)( 58,115)( 59,114)( 60,116)( 61, 97)( 62, 99)
( 63, 98)( 64,100)( 65,105)( 66,107)( 67,106)( 68,108)( 69,101)( 70,103)
( 71,102)( 72,104)( 73, 85)( 74, 87)( 75, 86)( 76, 88)( 77, 93)( 78, 95)
( 79, 94)( 80, 96)( 81, 89)( 82, 91)( 83, 90)( 84, 92)(157,317)(158,319)
(159,318)(160,320)(161,313)(162,315)(163,314)(164,316)(165,321)(166,323)
(167,322)(168,324)(169,461)(170,463)(171,462)(172,464)(173,457)(174,459)
(175,458)(176,460)(177,465)(178,467)(179,466)(180,468)(181,449)(182,451)
(183,450)(184,452)(185,445)(186,447)(187,446)(188,448)(189,453)(190,455)
(191,454)(192,456)(193,437)(194,439)(195,438)(196,440)(197,433)(198,435)
(199,434)(200,436)(201,441)(202,443)(203,442)(204,444)(205,425)(206,427)
(207,426)(208,428)(209,421)(210,423)(211,422)(212,424)(213,429)(214,431)
(215,430)(216,432)(217,413)(218,415)(219,414)(220,416)(221,409)(222,411)
(223,410)(224,412)(225,417)(226,419)(227,418)(228,420)(229,401)(230,403)
(231,402)(232,404)(233,397)(234,399)(235,398)(236,400)(237,405)(238,407)
(239,406)(240,408)(241,389)(242,391)(243,390)(244,392)(245,385)(246,387)
(247,386)(248,388)(249,393)(250,395)(251,394)(252,396)(253,377)(254,379)
(255,378)(256,380)(257,373)(258,375)(259,374)(260,376)(261,381)(262,383)
(263,382)(264,384)(265,365)(266,367)(267,366)(268,368)(269,361)(270,363)
(271,362)(272,364)(273,369)(274,371)(275,370)(276,372)(277,353)(278,355)
(279,354)(280,356)(281,349)(282,351)(283,350)(284,352)(285,357)(286,359)
(287,358)(288,360)(289,341)(290,343)(291,342)(292,344)(293,337)(294,339)
(295,338)(296,340)(297,345)(298,347)(299,346)(300,348)(301,329)(302,331)
(303,330)(304,332)(305,325)(306,327)(307,326)(308,328)(309,333)(310,335)
(311,334)(312,336);
s2 := Sym(468)!( 1,325)( 2,328)( 3,327)( 4,326)( 5,333)( 6,336)( 7,335)
( 8,334)( 9,329)( 10,332)( 11,331)( 12,330)( 13,313)( 14,316)( 15,315)
( 16,314)( 17,321)( 18,324)( 19,323)( 20,322)( 21,317)( 22,320)( 23,319)
( 24,318)( 25,457)( 26,460)( 27,459)( 28,458)( 29,465)( 30,468)( 31,467)
( 32,466)( 33,461)( 34,464)( 35,463)( 36,462)( 37,445)( 38,448)( 39,447)
( 40,446)( 41,453)( 42,456)( 43,455)( 44,454)( 45,449)( 46,452)( 47,451)
( 48,450)( 49,433)( 50,436)( 51,435)( 52,434)( 53,441)( 54,444)( 55,443)
( 56,442)( 57,437)( 58,440)( 59,439)( 60,438)( 61,421)( 62,424)( 63,423)
( 64,422)( 65,429)( 66,432)( 67,431)( 68,430)( 69,425)( 70,428)( 71,427)
( 72,426)( 73,409)( 74,412)( 75,411)( 76,410)( 77,417)( 78,420)( 79,419)
( 80,418)( 81,413)( 82,416)( 83,415)( 84,414)( 85,397)( 86,400)( 87,399)
( 88,398)( 89,405)( 90,408)( 91,407)( 92,406)( 93,401)( 94,404)( 95,403)
( 96,402)( 97,385)( 98,388)( 99,387)(100,386)(101,393)(102,396)(103,395)
(104,394)(105,389)(106,392)(107,391)(108,390)(109,373)(110,376)(111,375)
(112,374)(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)
(120,378)(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)
(128,370)(129,365)(130,368)(131,367)(132,366)(133,349)(134,352)(135,351)
(136,350)(137,357)(138,360)(139,359)(140,358)(141,353)(142,356)(143,355)
(144,354)(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)
(152,346)(153,341)(154,344)(155,343)(156,342)(157,169)(158,172)(159,171)
(160,170)(161,177)(162,180)(163,179)(164,178)(165,173)(166,176)(167,175)
(168,174)(181,301)(182,304)(183,303)(184,302)(185,309)(186,312)(187,311)
(188,310)(189,305)(190,308)(191,307)(192,306)(193,289)(194,292)(195,291)
(196,290)(197,297)(198,300)(199,299)(200,298)(201,293)(202,296)(203,295)
(204,294)(205,277)(206,280)(207,279)(208,278)(209,285)(210,288)(211,287)
(212,286)(213,281)(214,284)(215,283)(216,282)(217,265)(218,268)(219,267)
(220,266)(221,273)(222,276)(223,275)(224,274)(225,269)(226,272)(227,271)
(228,270)(229,253)(230,256)(231,255)(232,254)(233,261)(234,264)(235,263)
(236,262)(237,257)(238,260)(239,259)(240,258)(242,244)(245,249)(246,252)
(247,251)(248,250);
poly := sub<Sym(468)|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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope