Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944b
if this polytope has a name.
Group : SmallGroup(1944,942)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
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) :
   2-fold quotients : {18,3}*972a
   3-fold quotients : {6,6}*648c, {18,6}*648h
   6-fold quotients : {6,3}*324, {18,3}*324
   9-fold quotients : {6,6}*216a
   18-fold quotients : {6,3}*108
   27-fold quotients : {6,6}*72b
   54-fold quotients : {6,3}*36
   81-fold quotients : {2,6}*24
   162-fold quotients : {2,3}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)( 85,165)
( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)( 93,184)
( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)(101,177)
(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)(109,194)
(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)(117,198)
(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)(125,215)
(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)(133,205)
(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)(141,218)
(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)(149,235)
(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)(157,228)
(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)(249,250)
(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)(260,265)
(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)(283,296)
(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)(303,304)
(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)(314,319)
(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)(331,412)
(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)(339,425)
(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)(347,415)
(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)(355,435)
(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)(363,454)
(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)(371,447)
(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)(379,464)
(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)(387,468)
(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)(395,485)
(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)(403,475)
(404,476)(405,477);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10, 92)( 11, 91)( 12, 93)( 13, 98)( 14, 97)( 15, 99)( 16, 95)
( 17, 94)( 18, 96)( 19,102)( 20,101)( 21,100)( 22,108)( 23,107)( 24,106)
( 25,105)( 26,104)( 27,103)( 28,157)( 29,159)( 30,158)( 31,154)( 32,156)
( 33,155)( 34,160)( 35,162)( 36,161)( 37,140)( 38,139)( 39,141)( 40,137)
( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,150)( 47,149)( 48,148)
( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,122)( 56,121)
( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)( 64,132)
( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)( 72,133)
( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)( 80,117)
( 81,116)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)(174,175)
(178,180)(181,184)(182,186)(183,185)(188,189)(190,242)(191,241)(192,243)
(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,225)(200,224)
(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)(208,232)
(209,234)(210,233)(211,229)(212,231)(213,230)(214,226)(215,228)(216,227)
(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)(251,330)
(252,329)(253,335)(254,334)(255,336)(256,341)(257,340)(258,342)(259,338)
(260,337)(261,339)(262,345)(263,344)(264,343)(265,351)(266,350)(267,349)
(268,348)(269,347)(270,346)(271,400)(272,402)(273,401)(274,397)(275,399)
(276,398)(277,403)(278,405)(279,404)(280,383)(281,382)(282,384)(283,380)
(284,379)(285,381)(286,386)(287,385)(288,387)(289,393)(290,392)(291,391)
(292,390)(293,389)(294,388)(295,396)(296,395)(297,394)(298,365)(299,364)
(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)(307,375)
(308,374)(309,373)(310,372)(311,371)(312,370)(313,378)(314,377)(315,376)
(316,355)(317,357)(318,356)(319,352)(320,354)(321,353)(322,358)(323,360)
(324,359)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)(417,418)
(421,423)(424,427)(425,429)(426,428)(431,432)(433,485)(434,484)(435,486)
(436,482)(437,481)(438,483)(439,479)(440,478)(441,480)(442,468)(443,467)
(444,466)(445,465)(446,464)(447,463)(448,462)(449,461)(450,460)(451,475)
(452,477)(453,476)(454,472)(455,474)(456,473)(457,469)(458,471)(459,470);;
s2 := (  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)(  8,276)
(  9,275)( 10,288)( 11,287)( 12,286)( 13,285)( 14,284)( 15,283)( 16,282)
( 17,281)( 18,280)( 19,293)( 20,292)( 21,294)( 22,290)( 23,289)( 24,291)
( 25,296)( 26,295)( 27,297)( 28,244)( 29,246)( 30,245)( 31,250)( 32,252)
( 33,251)( 34,247)( 35,249)( 36,248)( 37,261)( 38,260)( 39,259)( 40,258)
( 41,257)( 42,256)( 43,255)( 44,254)( 45,253)( 46,266)( 47,265)( 48,267)
( 49,263)( 50,262)( 51,264)( 52,269)( 53,268)( 54,270)( 55,299)( 56,298)
( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)( 64,313)
( 65,315)( 66,314)( 67,310)( 68,312)( 69,311)( 70,307)( 71,309)( 72,308)
( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)( 80,323)
( 81,322)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)( 88,440)
( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)( 96,449)
( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)(104,455)
(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)(112,407)
(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)(120,416)
(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)(128,431)
(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)(136,465)
(137,464)(138,463)(139,462)(140,461)(141,460)(142,468)(143,467)(144,466)
(145,470)(146,469)(147,471)(148,476)(149,475)(150,477)(151,473)(152,472)
(153,474)(154,484)(155,486)(156,485)(157,481)(158,483)(159,482)(160,478)
(161,480)(162,479)(163,356)(164,355)(165,357)(166,353)(167,352)(168,354)
(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)(176,369)
(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)(184,375)
(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)(192,330)
(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)(200,336)
(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)(208,351)
(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)(216,343)
(217,384)(218,383)(219,382)(220,381)(221,380)(222,379)(223,387)(224,386)
(225,385)(226,389)(227,388)(228,390)(229,395)(230,394)(231,396)(232,392)
(233,391)(234,393)(235,403)(236,405)(237,404)(238,400)(239,402)(240,401)
(241,397)(242,399)(243,398);;
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*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(486)!(  4,  8)(  5,  9)(  6,  7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 82,167)( 83,168)( 84,166)
( 85,165)( 86,163)( 87,164)( 88,169)( 89,170)( 90,171)( 91,185)( 92,186)
( 93,184)( 94,183)( 95,181)( 96,182)( 97,187)( 98,188)( 99,189)(100,176)
(101,177)(102,175)(103,174)(104,172)(105,173)(106,178)(107,179)(108,180)
(109,194)(110,195)(111,193)(112,192)(113,190)(114,191)(115,196)(116,197)
(117,198)(118,212)(119,213)(120,211)(121,210)(122,208)(123,209)(124,214)
(125,215)(126,216)(127,203)(128,204)(129,202)(130,201)(131,199)(132,200)
(133,205)(134,206)(135,207)(136,221)(137,222)(138,220)(139,219)(140,217)
(141,218)(142,223)(143,224)(144,225)(145,239)(146,240)(147,238)(148,237)
(149,235)(150,236)(151,241)(152,242)(153,243)(154,230)(155,231)(156,229)
(157,228)(158,226)(159,227)(160,232)(161,233)(162,234)(247,251)(248,252)
(249,250)(253,262)(254,263)(255,264)(256,269)(257,270)(258,268)(259,267)
(260,265)(261,266)(274,278)(275,279)(276,277)(280,289)(281,290)(282,291)
(283,296)(284,297)(285,295)(286,294)(287,292)(288,293)(301,305)(302,306)
(303,304)(307,316)(308,317)(309,318)(310,323)(311,324)(312,322)(313,321)
(314,319)(315,320)(325,410)(326,411)(327,409)(328,408)(329,406)(330,407)
(331,412)(332,413)(333,414)(334,428)(335,429)(336,427)(337,426)(338,424)
(339,425)(340,430)(341,431)(342,432)(343,419)(344,420)(345,418)(346,417)
(347,415)(348,416)(349,421)(350,422)(351,423)(352,437)(353,438)(354,436)
(355,435)(356,433)(357,434)(358,439)(359,440)(360,441)(361,455)(362,456)
(363,454)(364,453)(365,451)(366,452)(367,457)(368,458)(369,459)(370,446)
(371,447)(372,445)(373,444)(374,442)(375,443)(376,448)(377,449)(378,450)
(379,464)(380,465)(381,463)(382,462)(383,460)(384,461)(385,466)(386,467)
(387,468)(388,482)(389,483)(390,481)(391,480)(392,478)(393,479)(394,484)
(395,485)(396,486)(397,473)(398,474)(399,472)(400,471)(401,469)(402,470)
(403,475)(404,476)(405,477);
s1 := Sym(486)!(  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)
(  8, 87)(  9, 86)( 10, 92)( 11, 91)( 12, 93)( 13, 98)( 14, 97)( 15, 99)
( 16, 95)( 17, 94)( 18, 96)( 19,102)( 20,101)( 21,100)( 22,108)( 23,107)
( 24,106)( 25,105)( 26,104)( 27,103)( 28,157)( 29,159)( 30,158)( 31,154)
( 32,156)( 33,155)( 34,160)( 35,162)( 36,161)( 37,140)( 38,139)( 39,141)
( 40,137)( 41,136)( 42,138)( 43,143)( 44,142)( 45,144)( 46,150)( 47,149)
( 48,148)( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,122)
( 56,121)( 57,123)( 58,119)( 59,118)( 60,120)( 61,125)( 62,124)( 63,126)
( 64,132)( 65,131)( 66,130)( 67,129)( 68,128)( 69,127)( 70,135)( 71,134)
( 72,133)( 73,112)( 74,114)( 75,113)( 76,109)( 77,111)( 78,110)( 79,115)
( 80,117)( 81,116)(163,167)(164,166)(165,168)(169,170)(172,177)(173,176)
(174,175)(178,180)(181,184)(182,186)(183,185)(188,189)(190,242)(191,241)
(192,243)(193,239)(194,238)(195,240)(196,236)(197,235)(198,237)(199,225)
(200,224)(201,223)(202,222)(203,221)(204,220)(205,219)(206,218)(207,217)
(208,232)(209,234)(210,233)(211,229)(212,231)(213,230)(214,226)(215,228)
(216,227)(244,325)(245,327)(246,326)(247,331)(248,333)(249,332)(250,328)
(251,330)(252,329)(253,335)(254,334)(255,336)(256,341)(257,340)(258,342)
(259,338)(260,337)(261,339)(262,345)(263,344)(264,343)(265,351)(266,350)
(267,349)(268,348)(269,347)(270,346)(271,400)(272,402)(273,401)(274,397)
(275,399)(276,398)(277,403)(278,405)(279,404)(280,383)(281,382)(282,384)
(283,380)(284,379)(285,381)(286,386)(287,385)(288,387)(289,393)(290,392)
(291,391)(292,390)(293,389)(294,388)(295,396)(296,395)(297,394)(298,365)
(299,364)(300,366)(301,362)(302,361)(303,363)(304,368)(305,367)(306,369)
(307,375)(308,374)(309,373)(310,372)(311,371)(312,370)(313,378)(314,377)
(315,376)(316,355)(317,357)(318,356)(319,352)(320,354)(321,353)(322,358)
(323,360)(324,359)(406,410)(407,409)(408,411)(412,413)(415,420)(416,419)
(417,418)(421,423)(424,427)(425,429)(426,428)(431,432)(433,485)(434,484)
(435,486)(436,482)(437,481)(438,483)(439,479)(440,478)(441,480)(442,468)
(443,467)(444,466)(445,465)(446,464)(447,463)(448,462)(449,461)(450,460)
(451,475)(452,477)(453,476)(454,472)(455,474)(456,473)(457,469)(458,471)
(459,470);
s2 := Sym(486)!(  1,271)(  2,273)(  3,272)(  4,277)(  5,279)(  6,278)(  7,274)
(  8,276)(  9,275)( 10,288)( 11,287)( 12,286)( 13,285)( 14,284)( 15,283)
( 16,282)( 17,281)( 18,280)( 19,293)( 20,292)( 21,294)( 22,290)( 23,289)
( 24,291)( 25,296)( 26,295)( 27,297)( 28,244)( 29,246)( 30,245)( 31,250)
( 32,252)( 33,251)( 34,247)( 35,249)( 36,248)( 37,261)( 38,260)( 39,259)
( 40,258)( 41,257)( 42,256)( 43,255)( 44,254)( 45,253)( 46,266)( 47,265)
( 48,267)( 49,263)( 50,262)( 51,264)( 52,269)( 53,268)( 54,270)( 55,299)
( 56,298)( 57,300)( 58,305)( 59,304)( 60,306)( 61,302)( 62,301)( 63,303)
( 64,313)( 65,315)( 66,314)( 67,310)( 68,312)( 69,311)( 70,307)( 71,309)
( 72,308)( 73,321)( 74,320)( 75,319)( 76,318)( 77,317)( 78,316)( 79,324)
( 80,323)( 81,322)( 82,437)( 83,436)( 84,438)( 85,434)( 86,433)( 87,435)
( 88,440)( 89,439)( 90,441)( 91,442)( 92,444)( 93,443)( 94,448)( 95,450)
( 96,449)( 97,445)( 98,447)( 99,446)(100,459)(101,458)(102,457)(103,456)
(104,455)(105,454)(106,453)(107,452)(108,451)(109,410)(110,409)(111,411)
(112,407)(113,406)(114,408)(115,413)(116,412)(117,414)(118,415)(119,417)
(120,416)(121,421)(122,423)(123,422)(124,418)(125,420)(126,419)(127,432)
(128,431)(129,430)(130,429)(131,428)(132,427)(133,426)(134,425)(135,424)
(136,465)(137,464)(138,463)(139,462)(140,461)(141,460)(142,468)(143,467)
(144,466)(145,470)(146,469)(147,471)(148,476)(149,475)(150,477)(151,473)
(152,472)(153,474)(154,484)(155,486)(156,485)(157,481)(158,483)(159,482)
(160,478)(161,480)(162,479)(163,356)(164,355)(165,357)(166,353)(167,352)
(168,354)(169,359)(170,358)(171,360)(172,361)(173,363)(174,362)(175,367)
(176,369)(177,368)(178,364)(179,366)(180,365)(181,378)(182,377)(183,376)
(184,375)(185,374)(186,373)(187,372)(188,371)(189,370)(190,329)(191,328)
(192,330)(193,326)(194,325)(195,327)(196,332)(197,331)(198,333)(199,334)
(200,336)(201,335)(202,340)(203,342)(204,341)(205,337)(206,339)(207,338)
(208,351)(209,350)(210,349)(211,348)(212,347)(213,346)(214,345)(215,344)
(216,343)(217,384)(218,383)(219,382)(220,381)(221,380)(222,379)(223,387)
(224,386)(225,385)(226,389)(227,388)(228,390)(229,395)(230,394)(231,396)
(232,392)(233,391)(234,393)(235,403)(236,405)(237,404)(238,400)(239,402)
(240,401)(241,397)(242,399)(243,398);
poly := sub<Sym(486)|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*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1 >;
References : None.
to this polytope