Polytope of Type {18,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30189)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 144, 432, 48
Order of s0s1s2 : 72
Order of s0s1s2s1 : 12
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 : {18,6}*864
3-fold quotients : {6,6}*576b
4-fold quotients : {9,6}*432
6-fold quotients : {6,6}*288b
8-fold quotients : {18,6}*216b
9-fold quotients : {6,6}*192b
12-fold quotients : {3,6}*144
16-fold quotients : {9,6}*108
18-fold quotients : {6,6}*96
24-fold quotients : {18,2}*72, {6,6}*72c
36-fold quotients : {3,6}*48, {6,3}*48
48-fold quotients : {9,2}*36, {3,6}*36
72-fold quotients : {3,3}*24, {6,2}*24
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,217)( 2,218)( 3,220)( 4,219)( 5,223)( 6,224)( 7,221)( 8,222)
( 9,233)( 10,234)( 11,236)( 12,235)( 13,239)( 14,240)( 15,237)( 16,238)
( 17,225)( 18,226)( 19,228)( 20,227)( 21,231)( 22,232)( 23,229)( 24,230)
( 25,281)( 26,282)( 27,284)( 28,283)( 29,287)( 30,288)( 31,285)( 32,286)
( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)( 40,278)
( 41,265)( 42,266)( 43,268)( 44,267)( 45,271)( 46,272)( 47,269)( 48,270)
( 49,257)( 50,258)( 51,260)( 52,259)( 53,263)( 54,264)( 55,261)( 56,262)
( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)( 64,254)
( 65,241)( 66,242)( 67,244)( 68,243)( 69,247)( 70,248)( 71,245)( 72,246)
( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)( 80,366)
( 81,377)( 82,378)( 83,380)( 84,379)( 85,383)( 86,384)( 87,381)( 88,382)
( 89,369)( 90,370)( 91,372)( 92,371)( 93,375)( 94,376)( 95,373)( 96,374)
( 97,425)( 98,426)( 99,428)(100,427)(101,431)(102,432)(103,429)(104,430)
(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)(112,422)
(113,409)(114,410)(115,412)(116,411)(117,415)(118,416)(119,413)(120,414)
(121,401)(122,402)(123,404)(124,403)(125,407)(126,408)(127,405)(128,406)
(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)(136,398)
(137,385)(138,386)(139,388)(140,387)(141,391)(142,392)(143,389)(144,390)
(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)(152,294)
(153,305)(154,306)(155,308)(156,307)(157,311)(158,312)(159,309)(160,310)
(161,297)(162,298)(163,300)(164,299)(165,303)(166,304)(167,301)(168,302)
(169,353)(170,354)(171,356)(172,355)(173,359)(174,360)(175,357)(176,358)
(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)(184,350)
(185,337)(186,338)(187,340)(188,339)(189,343)(190,344)(191,341)(192,342)
(193,329)(194,330)(195,332)(196,331)(197,335)(198,336)(199,333)(200,334)
(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)(208,326)
(209,313)(210,314)(211,316)(212,315)(213,319)(214,320)(215,317)(216,318);;
s1 := ( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,102)( 6,101)( 7, 99)( 8,100)
( 9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)( 16,116)
( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)( 24,108)
( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)( 32, 76)
( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)( 40, 92)
( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)( 48, 84)
( 49,137)( 50,138)( 51,143)( 52,144)( 53,142)( 54,141)( 55,139)( 56,140)
( 57,129)( 58,130)( 59,135)( 60,136)( 61,134)( 62,133)( 63,131)( 64,132)
( 65,121)( 66,122)( 67,127)( 68,128)( 69,126)( 70,125)( 71,123)( 72,124)
(145,169)(146,170)(147,175)(148,176)(149,174)(150,173)(151,171)(152,172)
(153,185)(154,186)(155,191)(156,192)(157,190)(158,189)(159,187)(160,188)
(161,177)(162,178)(163,183)(164,184)(165,182)(166,181)(167,179)(168,180)
(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)(200,212)
(203,207)(204,208)(205,206)(217,313)(218,314)(219,319)(220,320)(221,318)
(222,317)(223,315)(224,316)(225,329)(226,330)(227,335)(228,336)(229,334)
(230,333)(231,331)(232,332)(233,321)(234,322)(235,327)(236,328)(237,326)
(238,325)(239,323)(240,324)(241,289)(242,290)(243,295)(244,296)(245,294)
(246,293)(247,291)(248,292)(249,305)(250,306)(251,311)(252,312)(253,310)
(254,309)(255,307)(256,308)(257,297)(258,298)(259,303)(260,304)(261,302)
(262,301)(263,299)(264,300)(265,353)(266,354)(267,359)(268,360)(269,358)
(270,357)(271,355)(272,356)(273,345)(274,346)(275,351)(276,352)(277,350)
(278,349)(279,347)(280,348)(281,337)(282,338)(283,343)(284,344)(285,342)
(286,341)(287,339)(288,340)(361,385)(362,386)(363,391)(364,392)(365,390)
(366,389)(367,387)(368,388)(369,401)(370,402)(371,407)(372,408)(373,406)
(374,405)(375,403)(376,404)(377,393)(378,394)(379,399)(380,400)(381,398)
(382,397)(383,395)(384,396)(409,425)(410,426)(411,431)(412,432)(413,430)
(414,429)(415,427)(416,428)(419,423)(420,424)(421,422);;
s2 := ( 1, 3)( 2, 4)( 5, 6)( 9, 11)( 10, 12)( 13, 14)( 17, 19)( 18, 20)
( 21, 22)( 25, 27)( 26, 28)( 29, 30)( 33, 35)( 34, 36)( 37, 38)( 41, 43)
( 42, 44)( 45, 46)( 49, 51)( 50, 52)( 53, 54)( 57, 59)( 58, 60)( 61, 62)
( 65, 67)( 66, 68)( 69, 70)( 73,147)( 74,148)( 75,145)( 76,146)( 77,150)
( 78,149)( 79,151)( 80,152)( 81,155)( 82,156)( 83,153)( 84,154)( 85,158)
( 86,157)( 87,159)( 88,160)( 89,163)( 90,164)( 91,161)( 92,162)( 93,166)
( 94,165)( 95,167)( 96,168)( 97,171)( 98,172)( 99,169)(100,170)(101,174)
(102,173)(103,175)(104,176)(105,179)(106,180)(107,177)(108,178)(109,182)
(110,181)(111,183)(112,184)(113,187)(114,188)(115,185)(116,186)(117,190)
(118,189)(119,191)(120,192)(121,195)(122,196)(123,193)(124,194)(125,198)
(126,197)(127,199)(128,200)(129,203)(130,204)(131,201)(132,202)(133,206)
(134,205)(135,207)(136,208)(137,211)(138,212)(139,209)(140,210)(141,214)
(142,213)(143,215)(144,216)(217,220)(218,219)(223,224)(225,228)(226,227)
(231,232)(233,236)(234,235)(239,240)(241,244)(242,243)(247,248)(249,252)
(250,251)(255,256)(257,260)(258,259)(263,264)(265,268)(266,267)(271,272)
(273,276)(274,275)(279,280)(281,284)(282,283)(287,288)(289,364)(290,363)
(291,362)(292,361)(293,365)(294,366)(295,368)(296,367)(297,372)(298,371)
(299,370)(300,369)(301,373)(302,374)(303,376)(304,375)(305,380)(306,379)
(307,378)(308,377)(309,381)(310,382)(311,384)(312,383)(313,388)(314,387)
(315,386)(316,385)(317,389)(318,390)(319,392)(320,391)(321,396)(322,395)
(323,394)(324,393)(325,397)(326,398)(327,400)(328,399)(329,404)(330,403)
(331,402)(332,401)(333,405)(334,406)(335,408)(336,407)(337,412)(338,411)
(339,410)(340,409)(341,413)(342,414)(343,416)(344,415)(345,420)(346,419)
(347,418)(348,417)(349,421)(350,422)(351,424)(352,423)(353,428)(354,427)
(355,426)(356,425)(357,429)(358,430)(359,432)(360,431);;
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*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*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(432)!( 1,217)( 2,218)( 3,220)( 4,219)( 5,223)( 6,224)( 7,221)
( 8,222)( 9,233)( 10,234)( 11,236)( 12,235)( 13,239)( 14,240)( 15,237)
( 16,238)( 17,225)( 18,226)( 19,228)( 20,227)( 21,231)( 22,232)( 23,229)
( 24,230)( 25,281)( 26,282)( 27,284)( 28,283)( 29,287)( 30,288)( 31,285)
( 32,286)( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)
( 40,278)( 41,265)( 42,266)( 43,268)( 44,267)( 45,271)( 46,272)( 47,269)
( 48,270)( 49,257)( 50,258)( 51,260)( 52,259)( 53,263)( 54,264)( 55,261)
( 56,262)( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)
( 64,254)( 65,241)( 66,242)( 67,244)( 68,243)( 69,247)( 70,248)( 71,245)
( 72,246)( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)
( 80,366)( 81,377)( 82,378)( 83,380)( 84,379)( 85,383)( 86,384)( 87,381)
( 88,382)( 89,369)( 90,370)( 91,372)( 92,371)( 93,375)( 94,376)( 95,373)
( 96,374)( 97,425)( 98,426)( 99,428)(100,427)(101,431)(102,432)(103,429)
(104,430)(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)
(112,422)(113,409)(114,410)(115,412)(116,411)(117,415)(118,416)(119,413)
(120,414)(121,401)(122,402)(123,404)(124,403)(125,407)(126,408)(127,405)
(128,406)(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)
(136,398)(137,385)(138,386)(139,388)(140,387)(141,391)(142,392)(143,389)
(144,390)(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)
(152,294)(153,305)(154,306)(155,308)(156,307)(157,311)(158,312)(159,309)
(160,310)(161,297)(162,298)(163,300)(164,299)(165,303)(166,304)(167,301)
(168,302)(169,353)(170,354)(171,356)(172,355)(173,359)(174,360)(175,357)
(176,358)(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)
(184,350)(185,337)(186,338)(187,340)(188,339)(189,343)(190,344)(191,341)
(192,342)(193,329)(194,330)(195,332)(196,331)(197,335)(198,336)(199,333)
(200,334)(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)
(208,326)(209,313)(210,314)(211,316)(212,315)(213,319)(214,320)(215,317)
(216,318);
s1 := Sym(432)!( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,102)( 6,101)( 7, 99)
( 8,100)( 9,113)( 10,114)( 11,119)( 12,120)( 13,118)( 14,117)( 15,115)
( 16,116)( 17,105)( 18,106)( 19,111)( 20,112)( 21,110)( 22,109)( 23,107)
( 24,108)( 25, 73)( 26, 74)( 27, 79)( 28, 80)( 29, 78)( 30, 77)( 31, 75)
( 32, 76)( 33, 89)( 34, 90)( 35, 95)( 36, 96)( 37, 94)( 38, 93)( 39, 91)
( 40, 92)( 41, 81)( 42, 82)( 43, 87)( 44, 88)( 45, 86)( 46, 85)( 47, 83)
( 48, 84)( 49,137)( 50,138)( 51,143)( 52,144)( 53,142)( 54,141)( 55,139)
( 56,140)( 57,129)( 58,130)( 59,135)( 60,136)( 61,134)( 62,133)( 63,131)
( 64,132)( 65,121)( 66,122)( 67,127)( 68,128)( 69,126)( 70,125)( 71,123)
( 72,124)(145,169)(146,170)(147,175)(148,176)(149,174)(150,173)(151,171)
(152,172)(153,185)(154,186)(155,191)(156,192)(157,190)(158,189)(159,187)
(160,188)(161,177)(162,178)(163,183)(164,184)(165,182)(166,181)(167,179)
(168,180)(193,209)(194,210)(195,215)(196,216)(197,214)(198,213)(199,211)
(200,212)(203,207)(204,208)(205,206)(217,313)(218,314)(219,319)(220,320)
(221,318)(222,317)(223,315)(224,316)(225,329)(226,330)(227,335)(228,336)
(229,334)(230,333)(231,331)(232,332)(233,321)(234,322)(235,327)(236,328)
(237,326)(238,325)(239,323)(240,324)(241,289)(242,290)(243,295)(244,296)
(245,294)(246,293)(247,291)(248,292)(249,305)(250,306)(251,311)(252,312)
(253,310)(254,309)(255,307)(256,308)(257,297)(258,298)(259,303)(260,304)
(261,302)(262,301)(263,299)(264,300)(265,353)(266,354)(267,359)(268,360)
(269,358)(270,357)(271,355)(272,356)(273,345)(274,346)(275,351)(276,352)
(277,350)(278,349)(279,347)(280,348)(281,337)(282,338)(283,343)(284,344)
(285,342)(286,341)(287,339)(288,340)(361,385)(362,386)(363,391)(364,392)
(365,390)(366,389)(367,387)(368,388)(369,401)(370,402)(371,407)(372,408)
(373,406)(374,405)(375,403)(376,404)(377,393)(378,394)(379,399)(380,400)
(381,398)(382,397)(383,395)(384,396)(409,425)(410,426)(411,431)(412,432)
(413,430)(414,429)(415,427)(416,428)(419,423)(420,424)(421,422);
s2 := Sym(432)!( 1, 3)( 2, 4)( 5, 6)( 9, 11)( 10, 12)( 13, 14)( 17, 19)
( 18, 20)( 21, 22)( 25, 27)( 26, 28)( 29, 30)( 33, 35)( 34, 36)( 37, 38)
( 41, 43)( 42, 44)( 45, 46)( 49, 51)( 50, 52)( 53, 54)( 57, 59)( 58, 60)
( 61, 62)( 65, 67)( 66, 68)( 69, 70)( 73,147)( 74,148)( 75,145)( 76,146)
( 77,150)( 78,149)( 79,151)( 80,152)( 81,155)( 82,156)( 83,153)( 84,154)
( 85,158)( 86,157)( 87,159)( 88,160)( 89,163)( 90,164)( 91,161)( 92,162)
( 93,166)( 94,165)( 95,167)( 96,168)( 97,171)( 98,172)( 99,169)(100,170)
(101,174)(102,173)(103,175)(104,176)(105,179)(106,180)(107,177)(108,178)
(109,182)(110,181)(111,183)(112,184)(113,187)(114,188)(115,185)(116,186)
(117,190)(118,189)(119,191)(120,192)(121,195)(122,196)(123,193)(124,194)
(125,198)(126,197)(127,199)(128,200)(129,203)(130,204)(131,201)(132,202)
(133,206)(134,205)(135,207)(136,208)(137,211)(138,212)(139,209)(140,210)
(141,214)(142,213)(143,215)(144,216)(217,220)(218,219)(223,224)(225,228)
(226,227)(231,232)(233,236)(234,235)(239,240)(241,244)(242,243)(247,248)
(249,252)(250,251)(255,256)(257,260)(258,259)(263,264)(265,268)(266,267)
(271,272)(273,276)(274,275)(279,280)(281,284)(282,283)(287,288)(289,364)
(290,363)(291,362)(292,361)(293,365)(294,366)(295,368)(296,367)(297,372)
(298,371)(299,370)(300,369)(301,373)(302,374)(303,376)(304,375)(305,380)
(306,379)(307,378)(308,377)(309,381)(310,382)(311,384)(312,383)(313,388)
(314,387)(315,386)(316,385)(317,389)(318,390)(319,392)(320,391)(321,396)
(322,395)(323,394)(324,393)(325,397)(326,398)(327,400)(328,399)(329,404)
(330,403)(331,402)(332,401)(333,405)(334,406)(335,408)(336,407)(337,412)
(338,411)(339,410)(340,409)(341,413)(342,414)(343,416)(344,415)(345,420)
(346,419)(347,418)(348,417)(349,421)(350,422)(351,424)(352,423)(353,428)
(354,427)(355,426)(356,425)(357,429)(358,430)(359,432)(360,431);
poly := sub<Sym(432)|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*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*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