Polytope of Type {117,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {117,4}*936
if this polytope has a name.
Group : SmallGroup(936,58)
Rank : 3
Schlafli Type : {117,4}
Number of vertices, edges, etc : 117, 234, 4
Order of s₀s₁s₂ : 117
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {117,4,2} of size 1872
Vertex Figure Of :
   {2,117,4} of size 1872
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {39,4}*312
   13-fold quotients : {9,4}*72
   39-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {117,4}*1872, {234,4}*1872b, {234,4}*1872c
Permutation Representation (GAP) :

s0 := (  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);;
s1 := (  1,169)(  2,170)(  3,172)(  4,171)(  5,177)(  6,178)(  7,180)(  8,179)
(  9,173)( 10,174)( 11,176)( 12,175)( 13,157)( 14,158)( 15,160)( 16,159)
( 17,165)( 18,166)( 19,168)( 20,167)( 21,161)( 22,162)( 23,164)( 24,163)
( 25,301)( 26,302)( 27,304)( 28,303)( 29,309)( 30,310)( 31,312)( 32,311)
( 33,305)( 34,306)( 35,308)( 36,307)( 37,289)( 38,290)( 39,292)( 40,291)
( 41,297)( 42,298)( 43,300)( 44,299)( 45,293)( 46,294)( 47,296)( 48,295)
( 49,277)( 50,278)( 51,280)( 52,279)( 53,285)( 54,286)( 55,288)( 56,287)
( 57,281)( 58,282)( 59,284)( 60,283)( 61,265)( 62,266)( 63,268)( 64,267)
( 65,273)( 66,274)( 67,276)( 68,275)( 69,269)( 70,270)( 71,272)( 72,271)
( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)( 80,263)
( 81,257)( 82,258)( 83,260)( 84,259)( 85,241)( 86,242)( 87,244)( 88,243)
( 89,249)( 90,250)( 91,252)( 92,251)( 93,245)( 94,246)( 95,248)( 96,247)
( 97,229)( 98,230)( 99,232)(100,231)(101,237)(102,238)(103,240)(104,239)
(105,233)(106,234)(107,236)(108,235)(109,217)(110,218)(111,220)(112,219)
(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)(120,223)
(121,205)(122,206)(123,208)(124,207)(125,213)(126,214)(127,216)(128,215)
(129,209)(130,210)(131,212)(132,211)(133,193)(134,194)(135,196)(136,195)
(137,201)(138,202)(139,204)(140,203)(141,197)(142,198)(143,200)(144,199)
(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)(151,192)(152,191)
(153,185)(154,186)(155,188)(156,187)(313,329)(314,330)(315,332)(316,331)
(317,325)(318,326)(319,328)(320,327)(321,333)(322,334)(323,336)(324,335)
(337,461)(338,462)(339,464)(340,463)(341,457)(342,458)(343,460)(344,459)
(345,465)(346,466)(347,468)(348,467)(349,449)(350,450)(351,452)(352,451)
(353,445)(354,446)(355,448)(356,447)(357,453)(358,454)(359,456)(360,455)
(361,437)(362,438)(363,440)(364,439)(365,433)(366,434)(367,436)(368,435)
(369,441)(370,442)(371,444)(372,443)(373,425)(374,426)(375,428)(376,427)
(377,421)(378,422)(379,424)(380,423)(381,429)(382,430)(383,432)(384,431)
(385,413)(386,414)(387,416)(388,415)(389,409)(390,410)(391,412)(392,411)
(393,417)(394,418)(395,420)(396,419)(397,401)(398,402)(399,404)(400,403)
(407,408);;
s2 := (  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)(334,335)
(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)(350,351)
(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)(366,367)
(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)(382,383)
(385,388)(386,387)(389,392)(390,391)(393,396)(394,395)(397,400)(398,399)
(401,404)(402,403)(405,408)(406,407)(409,412)(410,411)(413,416)(414,415)
(417,420)(418,419)(421,424)(422,423)(425,428)(426,427)(429,432)(430,431)
(433,436)(434,435)(437,440)(438,439)(441,444)(442,443)(445,448)(446,447)
(449,452)(450,451)(453,456)(454,455)(457,460)(458,459)(461,464)(462,463)
(465,468)(466,467);;
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, 
s2*s1*s0*s2*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*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*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*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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := 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);
s1 := Sym(468)!(  1,169)(  2,170)(  3,172)(  4,171)(  5,177)(  6,178)(  7,180)
(  8,179)(  9,173)( 10,174)( 11,176)( 12,175)( 13,157)( 14,158)( 15,160)
( 16,159)( 17,165)( 18,166)( 19,168)( 20,167)( 21,161)( 22,162)( 23,164)
( 24,163)( 25,301)( 26,302)( 27,304)( 28,303)( 29,309)( 30,310)( 31,312)
( 32,311)( 33,305)( 34,306)( 35,308)( 36,307)( 37,289)( 38,290)( 39,292)
( 40,291)( 41,297)( 42,298)( 43,300)( 44,299)( 45,293)( 46,294)( 47,296)
( 48,295)( 49,277)( 50,278)( 51,280)( 52,279)( 53,285)( 54,286)( 55,288)
( 56,287)( 57,281)( 58,282)( 59,284)( 60,283)( 61,265)( 62,266)( 63,268)
( 64,267)( 65,273)( 66,274)( 67,276)( 68,275)( 69,269)( 70,270)( 71,272)
( 72,271)( 73,253)( 74,254)( 75,256)( 76,255)( 77,261)( 78,262)( 79,264)
( 80,263)( 81,257)( 82,258)( 83,260)( 84,259)( 85,241)( 86,242)( 87,244)
( 88,243)( 89,249)( 90,250)( 91,252)( 92,251)( 93,245)( 94,246)( 95,248)
( 96,247)( 97,229)( 98,230)( 99,232)(100,231)(101,237)(102,238)(103,240)
(104,239)(105,233)(106,234)(107,236)(108,235)(109,217)(110,218)(111,220)
(112,219)(113,225)(114,226)(115,228)(116,227)(117,221)(118,222)(119,224)
(120,223)(121,205)(122,206)(123,208)(124,207)(125,213)(126,214)(127,216)
(128,215)(129,209)(130,210)(131,212)(132,211)(133,193)(134,194)(135,196)
(136,195)(137,201)(138,202)(139,204)(140,203)(141,197)(142,198)(143,200)
(144,199)(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)(151,192)
(152,191)(153,185)(154,186)(155,188)(156,187)(313,329)(314,330)(315,332)
(316,331)(317,325)(318,326)(319,328)(320,327)(321,333)(322,334)(323,336)
(324,335)(337,461)(338,462)(339,464)(340,463)(341,457)(342,458)(343,460)
(344,459)(345,465)(346,466)(347,468)(348,467)(349,449)(350,450)(351,452)
(352,451)(353,445)(354,446)(355,448)(356,447)(357,453)(358,454)(359,456)
(360,455)(361,437)(362,438)(363,440)(364,439)(365,433)(366,434)(367,436)
(368,435)(369,441)(370,442)(371,444)(372,443)(373,425)(374,426)(375,428)
(376,427)(377,421)(378,422)(379,424)(380,423)(381,429)(382,430)(383,432)
(384,431)(385,413)(386,414)(387,416)(388,415)(389,409)(390,410)(391,412)
(392,411)(393,417)(394,418)(395,420)(396,419)(397,401)(398,402)(399,404)
(400,403)(407,408);
s2 := Sym(468)!(  1,  4)(  2,  3)(  5,  8)(  6,  7)(  9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)
(334,335)(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)
(350,351)(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)
(366,367)(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)
(382,383)(385,388)(386,387)(389,392)(390,391)(393,396)(394,395)(397,400)
(398,399)(401,404)(402,403)(405,408)(406,407)(409,412)(410,411)(413,416)
(414,415)(417,420)(418,419)(421,424)(422,423)(425,428)(426,427)(429,432)
(430,431)(433,436)(434,435)(437,440)(438,439)(441,444)(442,443)(445,448)
(446,447)(449,452)(450,451)(453,456)(454,455)(457,460)(458,459)(461,464)
(462,463)(465,468)(466,467);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s1*s0*s2*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*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*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*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*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 >;

References : None.
to this polytope