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# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*1944t
if this polytope has a name.
Group : SmallGroup(1944,950)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 54, 486, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
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 : {9,18}*972g
3-fold quotients : {18,6}*648e
6-fold quotients : {9,6}*324d
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)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 59)
( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 74)( 38, 73)( 39, 75)
( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)( 47, 64)
( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)( 82,172)
( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)( 90,179)
( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)( 98,171)
( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)(106,187)
(107,189)(108,188)(109,227)(110,226)(111,228)(112,230)(113,229)(114,231)
(115,233)(116,232)(117,234)(118,218)(119,217)(120,219)(121,221)(122,220)
(123,222)(124,224)(125,223)(126,225)(127,236)(128,235)(129,237)(130,239)
(131,238)(132,240)(133,242)(134,241)(135,243)(136,200)(137,199)(138,201)
(139,203)(140,202)(141,204)(142,206)(143,205)(144,207)(145,191)(146,190)
(147,192)(148,194)(149,193)(150,195)(151,197)(152,196)(153,198)(154,209)
(155,208)(156,210)(157,212)(158,211)(159,213)(160,215)(161,214)(162,216)
(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)(257,267)
(258,266)(259,268)(260,270)(261,269)(271,299)(272,298)(273,300)(274,302)
(275,301)(276,303)(277,305)(278,304)(279,306)(280,317)(281,316)(282,318)
(283,320)(284,319)(285,321)(286,323)(287,322)(288,324)(289,308)(290,307)
(291,309)(292,311)(293,310)(294,312)(295,314)(296,313)(297,315)(325,415)
(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)(333,422)
(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)(341,414)
(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)(349,430)
(350,432)(351,431)(352,470)(353,469)(354,471)(355,473)(356,472)(357,474)
(358,476)(359,475)(360,477)(361,461)(362,460)(363,462)(364,464)(365,463)
(366,465)(367,467)(368,466)(369,468)(370,479)(371,478)(372,480)(373,482)
(374,481)(375,483)(376,485)(377,484)(378,486)(379,443)(380,442)(381,444)
(382,446)(383,445)(384,447)(385,449)(386,448)(387,450)(388,434)(389,433)
(390,435)(391,437)(392,436)(393,438)(394,440)(395,439)(396,441)(397,452)
(398,451)(399,453)(400,455)(401,454)(402,456)(403,458)(404,457)(405,459);;
s1 := (  1,436)(  2,438)(  3,437)(  4,440)(  5,439)(  6,441)(  7,435)(  8,434)
(  9,433)( 10,454)( 11,456)( 12,455)( 13,458)( 14,457)( 15,459)( 16,453)
( 17,452)( 18,451)( 19,445)( 20,447)( 21,446)( 22,449)( 23,448)( 24,450)
( 25,444)( 26,443)( 27,442)( 28,414)( 29,413)( 30,412)( 31,406)( 32,408)
( 33,407)( 34,410)( 35,409)( 36,411)( 37,432)( 38,431)( 39,430)( 40,424)
( 41,426)( 42,425)( 43,428)( 44,427)( 45,429)( 46,423)( 47,422)( 48,421)
( 49,415)( 50,417)( 51,416)( 52,419)( 53,418)( 54,420)( 55,462)( 56,461)
( 57,460)( 58,463)( 59,465)( 60,464)( 61,467)( 62,466)( 63,468)( 64,480)
( 65,479)( 66,478)( 67,481)( 68,483)( 69,482)( 70,485)( 71,484)( 72,486)
( 73,471)( 74,470)( 75,469)( 76,472)( 77,474)( 78,473)( 79,476)( 80,475)
( 81,477)( 82,355)( 83,357)( 84,356)( 85,359)( 86,358)( 87,360)( 88,354)
( 89,353)( 90,352)( 91,373)( 92,375)( 93,374)( 94,377)( 95,376)( 96,378)
( 97,372)( 98,371)( 99,370)(100,364)(101,366)(102,365)(103,368)(104,367)
(105,369)(106,363)(107,362)(108,361)(109,333)(110,332)(111,331)(112,325)
(113,327)(114,326)(115,329)(116,328)(117,330)(118,351)(119,350)(120,349)
(121,343)(122,345)(123,344)(124,347)(125,346)(126,348)(127,342)(128,341)
(129,340)(130,334)(131,336)(132,335)(133,338)(134,337)(135,339)(136,381)
(137,380)(138,379)(139,382)(140,384)(141,383)(142,386)(143,385)(144,387)
(145,399)(146,398)(147,397)(148,400)(149,402)(150,401)(151,404)(152,403)
(153,405)(154,390)(155,389)(156,388)(157,391)(158,393)(159,392)(160,395)
(161,394)(162,396)(163,274)(164,276)(165,275)(166,278)(167,277)(168,279)
(169,273)(170,272)(171,271)(172,292)(173,294)(174,293)(175,296)(176,295)
(177,297)(178,291)(179,290)(180,289)(181,283)(182,285)(183,284)(184,287)
(185,286)(186,288)(187,282)(188,281)(189,280)(190,252)(191,251)(192,250)
(193,244)(194,246)(195,245)(196,248)(197,247)(198,249)(199,270)(200,269)
(201,268)(202,262)(203,264)(204,263)(205,266)(206,265)(207,267)(208,261)
(209,260)(210,259)(211,253)(212,255)(213,254)(214,257)(215,256)(216,258)
(217,300)(218,299)(219,298)(220,301)(221,303)(222,302)(223,305)(224,304)
(225,306)(226,318)(227,317)(228,316)(229,319)(230,321)(231,320)(232,323)
(233,322)(234,324)(235,309)(236,308)(237,307)(238,310)(239,312)(240,311)
(241,314)(242,313)(243,315);;
s2 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 82,172)( 83,173)( 84,174)( 85,178)
( 86,179)( 87,180)( 88,175)( 89,176)( 90,177)( 91,163)( 92,164)( 93,165)
( 94,169)( 95,170)( 96,171)( 97,166)( 98,167)( 99,168)(100,181)(101,182)
(102,183)(103,187)(104,188)(105,189)(106,184)(107,185)(108,186)(109,199)
(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)(117,204)
(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)(125,194)
(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)(133,211)
(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)(141,234)
(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)(149,224)
(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)(157,241)
(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)(249,252)
(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)(260,266)
(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)(283,295)
(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)(303,306)
(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)(314,320)
(315,321)(325,415)(326,416)(327,417)(328,421)(329,422)(330,423)(331,418)
(332,419)(333,420)(334,406)(335,407)(336,408)(337,412)(338,413)(339,414)
(340,409)(341,410)(342,411)(343,424)(344,425)(345,426)(346,430)(347,431)
(348,432)(349,427)(350,428)(351,429)(352,442)(353,443)(354,444)(355,448)
(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)(363,435)
(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)(371,452)
(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,469)
(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)(387,474)
(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)(395,464)
(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)
(404,482)(405,483);;
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(486)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 28, 56)( 29, 55)( 30, 57)
( 31, 59)( 32, 58)( 33, 60)( 34, 62)( 35, 61)( 36, 63)( 37, 74)( 38, 73)
( 39, 75)( 40, 77)( 41, 76)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 65)
( 47, 64)( 48, 66)( 49, 68)( 50, 67)( 51, 69)( 52, 71)( 53, 70)( 54, 72)
( 82,172)( 83,174)( 84,173)( 85,175)( 86,177)( 87,176)( 88,178)( 89,180)
( 90,179)( 91,163)( 92,165)( 93,164)( 94,166)( 95,168)( 96,167)( 97,169)
( 98,171)( 99,170)(100,181)(101,183)(102,182)(103,184)(104,186)(105,185)
(106,187)(107,189)(108,188)(109,227)(110,226)(111,228)(112,230)(113,229)
(114,231)(115,233)(116,232)(117,234)(118,218)(119,217)(120,219)(121,221)
(122,220)(123,222)(124,224)(125,223)(126,225)(127,236)(128,235)(129,237)
(130,239)(131,238)(132,240)(133,242)(134,241)(135,243)(136,200)(137,199)
(138,201)(139,203)(140,202)(141,204)(142,206)(143,205)(144,207)(145,191)
(146,190)(147,192)(148,194)(149,193)(150,195)(151,197)(152,196)(153,198)
(154,209)(155,208)(156,210)(157,212)(158,211)(159,213)(160,215)(161,214)
(162,216)(245,246)(248,249)(251,252)(253,262)(254,264)(255,263)(256,265)
(257,267)(258,266)(259,268)(260,270)(261,269)(271,299)(272,298)(273,300)
(274,302)(275,301)(276,303)(277,305)(278,304)(279,306)(280,317)(281,316)
(282,318)(283,320)(284,319)(285,321)(286,323)(287,322)(288,324)(289,308)
(290,307)(291,309)(292,311)(293,310)(294,312)(295,314)(296,313)(297,315)
(325,415)(326,417)(327,416)(328,418)(329,420)(330,419)(331,421)(332,423)
(333,422)(334,406)(335,408)(336,407)(337,409)(338,411)(339,410)(340,412)
(341,414)(342,413)(343,424)(344,426)(345,425)(346,427)(347,429)(348,428)
(349,430)(350,432)(351,431)(352,470)(353,469)(354,471)(355,473)(356,472)
(357,474)(358,476)(359,475)(360,477)(361,461)(362,460)(363,462)(364,464)
(365,463)(366,465)(367,467)(368,466)(369,468)(370,479)(371,478)(372,480)
(373,482)(374,481)(375,483)(376,485)(377,484)(378,486)(379,443)(380,442)
(381,444)(382,446)(383,445)(384,447)(385,449)(386,448)(387,450)(388,434)
(389,433)(390,435)(391,437)(392,436)(393,438)(394,440)(395,439)(396,441)
(397,452)(398,451)(399,453)(400,455)(401,454)(402,456)(403,458)(404,457)
(405,459);
s1 := Sym(486)!(  1,436)(  2,438)(  3,437)(  4,440)(  5,439)(  6,441)(  7,435)
(  8,434)(  9,433)( 10,454)( 11,456)( 12,455)( 13,458)( 14,457)( 15,459)
( 16,453)( 17,452)( 18,451)( 19,445)( 20,447)( 21,446)( 22,449)( 23,448)
( 24,450)( 25,444)( 26,443)( 27,442)( 28,414)( 29,413)( 30,412)( 31,406)
( 32,408)( 33,407)( 34,410)( 35,409)( 36,411)( 37,432)( 38,431)( 39,430)
( 40,424)( 41,426)( 42,425)( 43,428)( 44,427)( 45,429)( 46,423)( 47,422)
( 48,421)( 49,415)( 50,417)( 51,416)( 52,419)( 53,418)( 54,420)( 55,462)
( 56,461)( 57,460)( 58,463)( 59,465)( 60,464)( 61,467)( 62,466)( 63,468)
( 64,480)( 65,479)( 66,478)( 67,481)( 68,483)( 69,482)( 70,485)( 71,484)
( 72,486)( 73,471)( 74,470)( 75,469)( 76,472)( 77,474)( 78,473)( 79,476)
( 80,475)( 81,477)( 82,355)( 83,357)( 84,356)( 85,359)( 86,358)( 87,360)
( 88,354)( 89,353)( 90,352)( 91,373)( 92,375)( 93,374)( 94,377)( 95,376)
( 96,378)( 97,372)( 98,371)( 99,370)(100,364)(101,366)(102,365)(103,368)
(104,367)(105,369)(106,363)(107,362)(108,361)(109,333)(110,332)(111,331)
(112,325)(113,327)(114,326)(115,329)(116,328)(117,330)(118,351)(119,350)
(120,349)(121,343)(122,345)(123,344)(124,347)(125,346)(126,348)(127,342)
(128,341)(129,340)(130,334)(131,336)(132,335)(133,338)(134,337)(135,339)
(136,381)(137,380)(138,379)(139,382)(140,384)(141,383)(142,386)(143,385)
(144,387)(145,399)(146,398)(147,397)(148,400)(149,402)(150,401)(151,404)
(152,403)(153,405)(154,390)(155,389)(156,388)(157,391)(158,393)(159,392)
(160,395)(161,394)(162,396)(163,274)(164,276)(165,275)(166,278)(167,277)
(168,279)(169,273)(170,272)(171,271)(172,292)(173,294)(174,293)(175,296)
(176,295)(177,297)(178,291)(179,290)(180,289)(181,283)(182,285)(183,284)
(184,287)(185,286)(186,288)(187,282)(188,281)(189,280)(190,252)(191,251)
(192,250)(193,244)(194,246)(195,245)(196,248)(197,247)(198,249)(199,270)
(200,269)(201,268)(202,262)(203,264)(204,263)(205,266)(206,265)(207,267)
(208,261)(209,260)(210,259)(211,253)(212,255)(213,254)(214,257)(215,256)
(216,258)(217,300)(218,299)(219,298)(220,301)(221,303)(222,302)(223,305)
(224,304)(225,306)(226,318)(227,317)(228,316)(229,319)(230,321)(231,320)
(232,323)(233,322)(234,324)(235,309)(236,308)(237,307)(238,310)(239,312)
(240,311)(241,314)(242,313)(243,315);
s2 := Sym(486)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 82,172)( 83,173)( 84,174)
( 85,178)( 86,179)( 87,180)( 88,175)( 89,176)( 90,177)( 91,163)( 92,164)
( 93,165)( 94,169)( 95,170)( 96,171)( 97,166)( 98,167)( 99,168)(100,181)
(101,182)(102,183)(103,187)(104,188)(105,189)(106,184)(107,185)(108,186)
(109,199)(110,200)(111,201)(112,205)(113,206)(114,207)(115,202)(116,203)
(117,204)(118,190)(119,191)(120,192)(121,196)(122,197)(123,198)(124,193)
(125,194)(126,195)(127,208)(128,209)(129,210)(130,214)(131,215)(132,216)
(133,211)(134,212)(135,213)(136,226)(137,227)(138,228)(139,232)(140,233)
(141,234)(142,229)(143,230)(144,231)(145,217)(146,218)(147,219)(148,223)
(149,224)(150,225)(151,220)(152,221)(153,222)(154,235)(155,236)(156,237)
(157,241)(158,242)(159,243)(160,238)(161,239)(162,240)(247,250)(248,251)
(249,252)(253,262)(254,263)(255,264)(256,268)(257,269)(258,270)(259,265)
(260,266)(261,267)(274,277)(275,278)(276,279)(280,289)(281,290)(282,291)
(283,295)(284,296)(285,297)(286,292)(287,293)(288,294)(301,304)(302,305)
(303,306)(307,316)(308,317)(309,318)(310,322)(311,323)(312,324)(313,319)
(314,320)(315,321)(325,415)(326,416)(327,417)(328,421)(329,422)(330,423)
(331,418)(332,419)(333,420)(334,406)(335,407)(336,408)(337,412)(338,413)
(339,414)(340,409)(341,410)(342,411)(343,424)(344,425)(345,426)(346,430)
(347,431)(348,432)(349,427)(350,428)(351,429)(352,442)(353,443)(354,444)
(355,448)(356,449)(357,450)(358,445)(359,446)(360,447)(361,433)(362,434)
(363,435)(364,439)(365,440)(366,441)(367,436)(368,437)(369,438)(370,451)
(371,452)(372,453)(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)
(379,469)(380,470)(381,471)(382,475)(383,476)(384,477)(385,472)(386,473)
(387,474)(388,460)(389,461)(390,462)(391,466)(392,467)(393,468)(394,463)
(395,464)(396,465)(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)
(403,481)(404,482)(405,483);
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*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 >;

```
References : None.
to this polytope