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# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*1944k
if this polytope has a name.
Group : SmallGroup(1944,956)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 162, 486, 54
Order of s0s1s2 : 6
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 : {18,3}*972b
3-fold quotients : {18,6}*648h
6-fold quotients : {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 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 26)
( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 74)( 65, 73)
( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)( 82,166)
( 83,168)( 84,167)( 85,163)( 86,165)( 87,164)( 88,169)( 89,171)( 90,170)
( 91,185)( 92,184)( 93,186)( 94,182)( 95,181)( 96,183)( 97,188)( 98,187)
( 99,189)(100,176)(101,175)(102,177)(103,173)(104,172)(105,174)(106,179)
(107,178)(108,180)(109,195)(110,194)(111,193)(112,192)(113,191)(114,190)
(115,198)(116,197)(117,196)(118,211)(119,213)(120,212)(121,208)(122,210)
(123,209)(124,214)(125,216)(126,215)(127,202)(128,204)(129,203)(130,199)
(131,201)(132,200)(133,205)(134,207)(135,206)(136,221)(137,220)(138,222)
(139,218)(140,217)(141,219)(142,224)(143,223)(144,225)(145,240)(146,239)
(147,238)(148,237)(149,236)(150,235)(151,243)(152,242)(153,241)(154,231)
(155,230)(156,229)(157,228)(158,227)(159,226)(160,234)(161,233)(162,232)
(245,246)(247,250)(248,252)(249,251)(253,263)(254,262)(255,264)(256,269)
(257,268)(258,270)(259,266)(260,265)(261,267)(272,273)(274,277)(275,279)
(276,278)(280,290)(281,289)(282,291)(283,296)(284,295)(285,297)(286,293)
(287,292)(288,294)(299,300)(301,304)(302,306)(303,305)(307,317)(308,316)
(309,318)(310,323)(311,322)(312,324)(313,320)(314,319)(315,321)(325,409)
(326,411)(327,410)(328,406)(329,408)(330,407)(331,412)(332,414)(333,413)
(334,428)(335,427)(336,429)(337,425)(338,424)(339,426)(340,431)(341,430)
(342,432)(343,419)(344,418)(345,420)(346,416)(347,415)(348,417)(349,422)
(350,421)(351,423)(352,438)(353,437)(354,436)(355,435)(356,434)(357,433)
(358,441)(359,440)(360,439)(361,454)(362,456)(363,455)(364,451)(365,453)
(366,452)(367,457)(368,459)(369,458)(370,445)(371,447)(372,446)(373,442)
(374,444)(375,443)(376,448)(377,450)(378,449)(379,464)(380,463)(381,465)
(382,461)(383,460)(384,462)(385,467)(386,466)(387,468)(388,483)(389,482)
(390,481)(391,480)(392,479)(393,478)(394,486)(395,485)(396,484)(397,474)
(398,473)(399,472)(400,471)(401,470)(402,469)(403,477)(404,476)(405,475);;
s1 := (  1, 82)(  2, 83)(  3, 84)(  4, 88)(  5, 89)(  6, 90)(  7, 85)(  8, 86)
(  9, 87)( 10, 91)( 11, 92)( 12, 93)( 13, 97)( 14, 98)( 15, 99)( 16, 94)
( 17, 95)( 18, 96)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)( 24,108)
( 25,103)( 26,104)( 27,105)( 28,158)( 29,159)( 30,157)( 31,155)( 32,156)
( 33,154)( 34,161)( 35,162)( 36,160)( 37,139)( 38,140)( 39,141)( 40,136)
( 41,137)( 42,138)( 43,142)( 44,143)( 45,144)( 46,148)( 47,149)( 48,150)
( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,121)( 56,122)
( 57,123)( 58,118)( 59,119)( 60,120)( 61,124)( 62,125)( 63,126)( 64,130)
( 65,131)( 66,132)( 67,127)( 68,128)( 69,129)( 70,133)( 71,134)( 72,135)
( 73,114)( 74,112)( 75,113)( 76,111)( 77,109)( 78,110)( 79,117)( 80,115)
( 81,116)(163,166)(164,167)(165,168)(172,175)(173,176)(174,177)(181,184)
(182,185)(183,186)(190,243)(191,241)(192,242)(193,240)(194,238)(195,239)
(196,237)(197,235)(198,236)(199,224)(200,225)(201,223)(202,221)(203,222)
(204,220)(205,218)(206,219)(207,217)(208,233)(209,234)(210,232)(211,230)
(212,231)(213,229)(214,227)(215,228)(216,226)(244,325)(245,326)(246,327)
(247,331)(248,332)(249,333)(250,328)(251,329)(252,330)(253,334)(254,335)
(255,336)(256,340)(257,341)(258,342)(259,337)(260,338)(261,339)(262,343)
(263,344)(264,345)(265,349)(266,350)(267,351)(268,346)(269,347)(270,348)
(271,401)(272,402)(273,400)(274,398)(275,399)(276,397)(277,404)(278,405)
(279,403)(280,382)(281,383)(282,384)(283,379)(284,380)(285,381)(286,385)
(287,386)(288,387)(289,391)(290,392)(291,393)(292,388)(293,389)(294,390)
(295,394)(296,395)(297,396)(298,364)(299,365)(300,366)(301,361)(302,362)
(303,363)(304,367)(305,368)(306,369)(307,373)(308,374)(309,375)(310,370)
(311,371)(312,372)(313,376)(314,377)(315,378)(316,357)(317,355)(318,356)
(319,354)(320,352)(321,353)(322,360)(323,358)(324,359)(406,409)(407,410)
(408,411)(415,418)(416,419)(417,420)(424,427)(425,428)(426,429)(433,486)
(434,484)(435,485)(436,483)(437,481)(438,482)(439,480)(440,478)(441,479)
(442,467)(443,468)(444,466)(445,464)(446,465)(447,463)(448,461)(449,462)
(450,460)(451,476)(452,477)(453,475)(454,473)(455,474)(456,472)(457,470)
(458,471)(459,469);;
s2 := (  1,271)(  2,272)(  3,273)(  4,278)(  5,279)(  6,277)(  7,276)(  8,274)
(  9,275)( 10,288)( 11,286)( 12,287)( 13,283)( 14,284)( 15,285)( 16,281)
( 17,282)( 18,280)( 19,293)( 20,294)( 21,292)( 22,291)( 23,289)( 24,290)
( 25,295)( 26,296)( 27,297)( 28,244)( 29,245)( 30,246)( 31,251)( 32,252)
( 33,250)( 34,249)( 35,247)( 36,248)( 37,261)( 38,259)( 39,260)( 40,256)
( 41,257)( 42,258)( 43,254)( 44,255)( 45,253)( 46,266)( 47,267)( 48,265)
( 49,264)( 50,262)( 51,263)( 52,268)( 53,269)( 54,270)( 55,298)( 56,299)
( 57,300)( 58,305)( 59,306)( 60,304)( 61,303)( 62,301)( 63,302)( 64,315)
( 65,313)( 66,314)( 67,310)( 68,311)( 69,312)( 70,308)( 71,309)( 72,307)
( 73,320)( 74,321)( 75,319)( 76,318)( 77,316)( 78,317)( 79,322)( 80,323)
( 81,324)( 82,438)( 83,436)( 84,437)( 85,433)( 86,434)( 87,435)( 88,440)
( 89,441)( 90,439)( 91,443)( 92,444)( 93,442)( 94,450)( 95,448)( 96,449)
( 97,445)( 98,446)( 99,447)(100,457)(101,458)(102,459)(103,455)(104,456)
(105,454)(106,453)(107,451)(108,452)(109,409)(110,410)(111,411)(112,407)
(113,408)(114,406)(115,414)(116,412)(117,413)(118,417)(119,415)(120,416)
(121,421)(122,422)(123,423)(124,419)(125,420)(126,418)(127,431)(128,432)
(129,430)(130,429)(131,427)(132,428)(133,424)(134,425)(135,426)(136,464)
(137,465)(138,463)(139,462)(140,460)(141,461)(142,466)(143,467)(144,468)
(145,469)(146,470)(147,471)(148,476)(149,477)(150,475)(151,474)(152,472)
(153,473)(154,486)(155,484)(156,485)(157,481)(158,482)(159,483)(160,479)
(161,480)(162,478)(163,357)(164,355)(165,356)(166,352)(167,353)(168,354)
(169,359)(170,360)(171,358)(172,362)(173,363)(174,361)(175,369)(176,367)
(177,368)(178,364)(179,365)(180,366)(181,376)(182,377)(183,378)(184,374)
(185,375)(186,373)(187,372)(188,370)(189,371)(190,328)(191,329)(192,330)
(193,326)(194,327)(195,325)(196,333)(197,331)(198,332)(199,336)(200,334)
(201,335)(202,340)(203,341)(204,342)(205,338)(206,339)(207,337)(208,350)
(209,351)(210,349)(211,348)(212,346)(213,347)(214,343)(215,344)(216,345)
(217,383)(218,384)(219,382)(220,381)(221,379)(222,380)(223,385)(224,386)
(225,387)(226,388)(227,389)(228,390)(229,395)(230,396)(231,394)(232,393)
(233,391)(234,392)(235,405)(236,403)(237,404)(238,400)(239,401)(240,402)
(241,398)(242,399)(243,397);;
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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope