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# Polytope of Type {6,18}

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

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

```
References : None.
to this polytope