Polytope of Type {18,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1944f
if this polytope has a name.
Group : SmallGroup(1944,944)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
2-fold quotients : {9,6}*972c
3-fold quotients : {18,6}*648d, {6,6}*648d
6-fold quotients : {9,6}*324b, {3,6}*324
9-fold quotients : {6,6}*216c
18-fold quotients : {3,6}*108
27-fold quotients : {6,6}*72c
54-fold quotients : {3,6}*36
81-fold quotients : {6,2}*24
162-fold quotients : {3,2}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)
( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 82,163)
( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)( 90,167)
( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)( 98,177)
( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)(106,184)
(107,186)(108,185)(109,218)(110,217)(111,219)(112,224)(113,223)(114,225)
(115,221)(116,220)(117,222)(118,227)(119,226)(120,228)(121,233)(122,232)
(123,234)(124,230)(125,229)(126,231)(127,236)(128,235)(129,237)(130,242)
(131,241)(132,243)(133,239)(134,238)(135,240)(136,191)(137,190)(138,192)
(139,197)(140,196)(141,198)(142,194)(143,193)(144,195)(145,200)(146,199)
(147,201)(148,206)(149,205)(150,207)(151,203)(152,202)(153,204)(154,209)
(155,208)(156,210)(157,215)(158,214)(159,216)(160,212)(161,211)(162,213)
(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)(258,260)
(263,264)(265,268)(266,270)(267,269)(271,299)(272,298)(273,300)(274,305)
(275,304)(276,306)(277,302)(278,301)(279,303)(280,308)(281,307)(282,309)
(283,314)(284,313)(285,315)(286,311)(287,310)(288,312)(289,317)(290,316)
(291,318)(292,323)(293,322)(294,324)(295,320)(296,319)(297,321)(325,406)
(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)(332,411)(333,410)
(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)(340,418)(341,420)
(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)(349,427)
(350,429)(351,428)(352,461)(353,460)(354,462)(355,467)(356,466)(357,468)
(358,464)(359,463)(360,465)(361,470)(362,469)(363,471)(364,476)(365,475)
(366,477)(367,473)(368,472)(369,474)(370,479)(371,478)(372,480)(373,485)
(374,484)(375,486)(376,482)(377,481)(378,483)(379,434)(380,433)(381,435)
(382,440)(383,439)(384,441)(385,437)(386,436)(387,438)(388,443)(389,442)
(390,444)(391,449)(392,448)(393,450)(394,446)(395,445)(396,447)(397,452)
(398,451)(399,453)(400,458)(401,457)(402,459)(403,455)(404,454)(405,456);;
s1 := ( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)( 8,447)
( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)( 16,452)
( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)( 24,433)
( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)( 32,425)
( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)( 40,412)
( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)( 48,423)
( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,460)( 56,462)
( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)( 64,476)
( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)( 72,471)
( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,478)( 79,486)( 80,485)
( 81,484)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)( 88,364)
( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)( 96,375)
( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)(104,353)
(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)(112,345)
(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)(120,326)
(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)(128,340)
(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)(136,379)
(137,381)(138,380)(139,385)(140,387)(141,386)(142,382)(143,384)(144,383)
(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)(152,388)
(153,390)(154,402)(155,401)(156,400)(157,399)(158,398)(159,397)(160,405)
(161,404)(162,403)(163,280)(164,282)(165,281)(166,286)(167,288)(168,287)
(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)(176,292)
(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)(184,273)
(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)(192,265)
(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)(200,246)
(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)(208,260)
(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)(216,255)
(217,298)(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)
(225,302)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)(232,308)
(233,307)(234,309)(235,321)(236,320)(237,319)(238,318)(239,317)(240,316)
(241,324)(242,323)(243,322);;
s2 := ( 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)( 82,163)( 83,164)( 84,165)( 85,170)
( 86,171)( 87,169)( 88,168)( 89,166)( 90,167)( 91,181)( 92,182)( 93,183)
( 94,188)( 95,189)( 96,187)( 97,186)( 98,184)( 99,185)(100,172)(101,173)
(102,174)(103,179)(104,180)(105,178)(106,177)(107,175)(108,176)(109,190)
(110,191)(111,192)(112,197)(113,198)(114,196)(115,195)(116,193)(117,194)
(118,208)(119,209)(120,210)(121,215)(122,216)(123,214)(124,213)(125,211)
(126,212)(127,199)(128,200)(129,201)(130,206)(131,207)(132,205)(133,204)
(134,202)(135,203)(136,217)(137,218)(138,219)(139,224)(140,225)(141,223)
(142,222)(143,220)(144,221)(145,235)(146,236)(147,237)(148,242)(149,243)
(150,241)(151,240)(152,238)(153,239)(154,226)(155,227)(156,228)(157,233)
(158,234)(159,232)(160,231)(161,229)(162,230)(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)(325,406)(326,407)(327,408)(328,413)(329,414)(330,412)(331,411)
(332,409)(333,410)(334,424)(335,425)(336,426)(337,431)(338,432)(339,430)
(340,429)(341,427)(342,428)(343,415)(344,416)(345,417)(346,422)(347,423)
(348,421)(349,420)(350,418)(351,419)(352,433)(353,434)(354,435)(355,440)
(356,441)(357,439)(358,438)(359,436)(360,437)(361,451)(362,452)(363,453)
(364,458)(365,459)(366,457)(367,456)(368,454)(369,455)(370,442)(371,443)
(372,444)(373,449)(374,450)(375,448)(376,447)(377,445)(378,446)(379,460)
(380,461)(381,462)(382,467)(383,468)(384,466)(385,465)(386,463)(387,464)
(388,478)(389,479)(390,480)(391,485)(392,486)(393,484)(394,483)(395,481)
(396,482)(397,469)(398,470)(399,471)(400,476)(401,477)(402,475)(403,474)
(404,472)(405,473);;
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*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(486)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 56)( 29, 55)( 30, 57)
( 31, 62)( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)
( 39, 66)( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)
( 47, 73)( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)
( 82,163)( 83,165)( 84,164)( 85,169)( 86,171)( 87,170)( 88,166)( 89,168)
( 90,167)( 91,172)( 92,174)( 93,173)( 94,178)( 95,180)( 96,179)( 97,175)
( 98,177)( 99,176)(100,181)(101,183)(102,182)(103,187)(104,189)(105,188)
(106,184)(107,186)(108,185)(109,218)(110,217)(111,219)(112,224)(113,223)
(114,225)(115,221)(116,220)(117,222)(118,227)(119,226)(120,228)(121,233)
(122,232)(123,234)(124,230)(125,229)(126,231)(127,236)(128,235)(129,237)
(130,242)(131,241)(132,243)(133,239)(134,238)(135,240)(136,191)(137,190)
(138,192)(139,197)(140,196)(141,198)(142,194)(143,193)(144,195)(145,200)
(146,199)(147,201)(148,206)(149,205)(150,207)(151,203)(152,202)(153,204)
(154,209)(155,208)(156,210)(157,215)(158,214)(159,216)(160,212)(161,211)
(162,213)(245,246)(247,250)(248,252)(249,251)(254,255)(256,259)(257,261)
(258,260)(263,264)(265,268)(266,270)(267,269)(271,299)(272,298)(273,300)
(274,305)(275,304)(276,306)(277,302)(278,301)(279,303)(280,308)(281,307)
(282,309)(283,314)(284,313)(285,315)(286,311)(287,310)(288,312)(289,317)
(290,316)(291,318)(292,323)(293,322)(294,324)(295,320)(296,319)(297,321)
(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)(332,411)
(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)(340,418)
(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)(348,431)
(349,427)(350,429)(351,428)(352,461)(353,460)(354,462)(355,467)(356,466)
(357,468)(358,464)(359,463)(360,465)(361,470)(362,469)(363,471)(364,476)
(365,475)(366,477)(367,473)(368,472)(369,474)(370,479)(371,478)(372,480)
(373,485)(374,484)(375,486)(376,482)(377,481)(378,483)(379,434)(380,433)
(381,435)(382,440)(383,439)(384,441)(385,437)(386,436)(387,438)(388,443)
(389,442)(390,444)(391,449)(392,448)(393,450)(394,446)(395,445)(396,447)
(397,452)(398,451)(399,453)(400,458)(401,457)(402,459)(403,455)(404,454)
(405,456);
s1 := Sym(486)!( 1,442)( 2,444)( 3,443)( 4,448)( 5,450)( 6,449)( 7,445)
( 8,447)( 9,446)( 10,458)( 11,457)( 12,459)( 13,455)( 14,454)( 15,456)
( 16,452)( 17,451)( 18,453)( 19,438)( 20,437)( 21,436)( 22,435)( 23,434)
( 24,433)( 25,441)( 26,440)( 27,439)( 28,429)( 29,428)( 30,427)( 31,426)
( 32,425)( 33,424)( 34,432)( 35,431)( 36,430)( 37,406)( 38,408)( 39,407)
( 40,412)( 41,414)( 42,413)( 43,409)( 44,411)( 45,410)( 46,422)( 47,421)
( 48,423)( 49,419)( 50,418)( 51,420)( 52,416)( 53,415)( 54,417)( 55,460)
( 56,462)( 57,461)( 58,466)( 59,468)( 60,467)( 61,463)( 62,465)( 63,464)
( 64,476)( 65,475)( 66,477)( 67,473)( 68,472)( 69,474)( 70,470)( 71,469)
( 72,471)( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,478)( 79,486)
( 80,485)( 81,484)( 82,361)( 83,363)( 84,362)( 85,367)( 86,369)( 87,368)
( 88,364)( 89,366)( 90,365)( 91,377)( 92,376)( 93,378)( 94,374)( 95,373)
( 96,375)( 97,371)( 98,370)( 99,372)(100,357)(101,356)(102,355)(103,354)
(104,353)(105,352)(106,360)(107,359)(108,358)(109,348)(110,347)(111,346)
(112,345)(113,344)(114,343)(115,351)(116,350)(117,349)(118,325)(119,327)
(120,326)(121,331)(122,333)(123,332)(124,328)(125,330)(126,329)(127,341)
(128,340)(129,342)(130,338)(131,337)(132,339)(133,335)(134,334)(135,336)
(136,379)(137,381)(138,380)(139,385)(140,387)(141,386)(142,382)(143,384)
(144,383)(145,395)(146,394)(147,396)(148,392)(149,391)(150,393)(151,389)
(152,388)(153,390)(154,402)(155,401)(156,400)(157,399)(158,398)(159,397)
(160,405)(161,404)(162,403)(163,280)(164,282)(165,281)(166,286)(167,288)
(168,287)(169,283)(170,285)(171,284)(172,296)(173,295)(174,297)(175,293)
(176,292)(177,294)(178,290)(179,289)(180,291)(181,276)(182,275)(183,274)
(184,273)(185,272)(186,271)(187,279)(188,278)(189,277)(190,267)(191,266)
(192,265)(193,264)(194,263)(195,262)(196,270)(197,269)(198,268)(199,244)
(200,246)(201,245)(202,250)(203,252)(204,251)(205,247)(206,249)(207,248)
(208,260)(209,259)(210,261)(211,257)(212,256)(213,258)(214,254)(215,253)
(216,255)(217,298)(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)
(224,303)(225,302)(226,314)(227,313)(228,315)(229,311)(230,310)(231,312)
(232,308)(233,307)(234,309)(235,321)(236,320)(237,319)(238,318)(239,317)
(240,316)(241,324)(242,323)(243,322);
s2 := 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)( 82,163)( 83,164)( 84,165)
( 85,170)( 86,171)( 87,169)( 88,168)( 89,166)( 90,167)( 91,181)( 92,182)
( 93,183)( 94,188)( 95,189)( 96,187)( 97,186)( 98,184)( 99,185)(100,172)
(101,173)(102,174)(103,179)(104,180)(105,178)(106,177)(107,175)(108,176)
(109,190)(110,191)(111,192)(112,197)(113,198)(114,196)(115,195)(116,193)
(117,194)(118,208)(119,209)(120,210)(121,215)(122,216)(123,214)(124,213)
(125,211)(126,212)(127,199)(128,200)(129,201)(130,206)(131,207)(132,205)
(133,204)(134,202)(135,203)(136,217)(137,218)(138,219)(139,224)(140,225)
(141,223)(142,222)(143,220)(144,221)(145,235)(146,236)(147,237)(148,242)
(149,243)(150,241)(151,240)(152,238)(153,239)(154,226)(155,227)(156,228)
(157,233)(158,234)(159,232)(160,231)(161,229)(162,230)(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)(325,406)(326,407)(327,408)(328,413)(329,414)(330,412)
(331,411)(332,409)(333,410)(334,424)(335,425)(336,426)(337,431)(338,432)
(339,430)(340,429)(341,427)(342,428)(343,415)(344,416)(345,417)(346,422)
(347,423)(348,421)(349,420)(350,418)(351,419)(352,433)(353,434)(354,435)
(355,440)(356,441)(357,439)(358,438)(359,436)(360,437)(361,451)(362,452)
(363,453)(364,458)(365,459)(366,457)(367,456)(368,454)(369,455)(370,442)
(371,443)(372,444)(373,449)(374,450)(375,448)(376,447)(377,445)(378,446)
(379,460)(380,461)(381,462)(382,467)(383,468)(384,466)(385,465)(386,463)
(387,464)(388,478)(389,479)(390,480)(391,485)(392,486)(393,484)(394,483)
(395,481)(396,482)(397,469)(398,470)(399,471)(400,476)(401,477)(402,475)
(403,474)(404,472)(405,473);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;
References : None.
to this polytope