Polytope of Type {18,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30313)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 72, 432, 48
Order of s0s1s2 : 36
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 : {18,12}*864b
3-fold quotients : {18,4}*576b, {6,12}*576f
4-fold quotients : {18,12}*432b, {9,12}*432
6-fold quotients : {18,4}*288, {6,12}*288b
8-fold quotients : {18,6}*216b
9-fold quotients : {6,4}*192b
12-fold quotients : {18,4}*144a, {9,4}*144, {18,4}*144b, {18,4}*144c, {6,12}*144c, {3,12}*144
16-fold quotients : {9,6}*108
18-fold quotients : {6,4}*96
24-fold quotients : {9,4}*72, {18,2}*72, {6,6}*72c
36-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {9,2}*36, {3,6}*36
72-fold quotients : {3,4}*24, {6,2}*24
108-fold quotients : {2,4}*16
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 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 25)( 14, 26)( 15, 28)
( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)( 23, 32)
( 24, 31)( 37, 81)( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)( 43, 80)
( 44, 79)( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49,105)( 50,106)( 51,108)
( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)( 59,100)
( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)( 67, 92)
( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(111,112)(113,117)(114,118)
(115,120)(116,119)(121,133)(122,134)(123,136)(124,135)(125,141)(126,142)
(127,144)(128,143)(129,137)(130,138)(131,140)(132,139)(145,189)(146,190)
(147,192)(148,191)(149,185)(150,186)(151,188)(152,187)(153,181)(154,182)
(155,184)(156,183)(157,213)(158,214)(159,216)(160,215)(161,209)(162,210)
(163,212)(164,211)(165,205)(166,206)(167,208)(168,207)(169,201)(170,202)
(171,204)(172,203)(173,197)(174,198)(175,200)(176,199)(177,193)(178,194)
(179,196)(180,195)(219,220)(221,225)(222,226)(223,228)(224,227)(229,241)
(230,242)(231,244)(232,243)(233,249)(234,250)(235,252)(236,251)(237,245)
(238,246)(239,248)(240,247)(253,297)(254,298)(255,300)(256,299)(257,293)
(258,294)(259,296)(260,295)(261,289)(262,290)(263,292)(264,291)(265,321)
(266,322)(267,324)(268,323)(269,317)(270,318)(271,320)(272,319)(273,313)
(274,314)(275,316)(276,315)(277,309)(278,310)(279,312)(280,311)(281,305)
(282,306)(283,308)(284,307)(285,301)(286,302)(287,304)(288,303)(327,328)
(329,333)(330,334)(331,336)(332,335)(337,349)(338,350)(339,352)(340,351)
(341,357)(342,358)(343,360)(344,359)(345,353)(346,354)(347,356)(348,355)
(361,405)(362,406)(363,408)(364,407)(365,401)(366,402)(367,404)(368,403)
(369,397)(370,398)(371,400)(372,399)(373,429)(374,430)(375,432)(376,431)
(377,425)(378,426)(379,428)(380,427)(381,421)(382,422)(383,424)(384,423)
(385,417)(386,418)(387,420)(388,419)(389,413)(390,414)(391,416)(392,415)
(393,409)(394,410)(395,412)(396,411);;
s1 := ( 1,265)( 2,268)( 3,267)( 4,266)( 5,273)( 6,276)( 7,275)( 8,274)
( 9,269)( 10,272)( 11,271)( 12,270)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)( 24,258)
( 25,277)( 26,280)( 27,279)( 28,278)( 29,285)( 30,288)( 31,287)( 32,286)
( 33,281)( 34,284)( 35,283)( 36,282)( 37,229)( 38,232)( 39,231)( 40,230)
( 41,237)( 42,240)( 43,239)( 44,238)( 45,233)( 46,236)( 47,235)( 48,234)
( 49,217)( 50,220)( 51,219)( 52,218)( 53,225)( 54,228)( 55,227)( 56,226)
( 57,221)( 58,224)( 59,223)( 60,222)( 61,241)( 62,244)( 63,243)( 64,242)
( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)( 72,246)
( 73,309)( 74,312)( 75,311)( 76,310)( 77,305)( 78,308)( 79,307)( 80,306)
( 81,301)( 82,304)( 83,303)( 84,302)( 85,297)( 86,300)( 87,299)( 88,298)
( 89,293)( 90,296)( 91,295)( 92,294)( 93,289)( 94,292)( 95,291)( 96,290)
( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)(104,318)
(105,313)(106,316)(107,315)(108,314)(109,373)(110,376)(111,375)(112,374)
(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)(120,378)
(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)(128,370)
(129,365)(130,368)(131,367)(132,366)(133,385)(134,388)(135,387)(136,386)
(137,393)(138,396)(139,395)(140,394)(141,389)(142,392)(143,391)(144,390)
(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)(152,346)
(153,341)(154,344)(155,343)(156,342)(157,325)(158,328)(159,327)(160,326)
(161,333)(162,336)(163,335)(164,334)(165,329)(166,332)(167,331)(168,330)
(169,349)(170,352)(171,351)(172,350)(173,357)(174,360)(175,359)(176,358)
(177,353)(178,356)(179,355)(180,354)(181,417)(182,420)(183,419)(184,418)
(185,413)(186,416)(187,415)(188,414)(189,409)(190,412)(191,411)(192,410)
(193,405)(194,408)(195,407)(196,406)(197,401)(198,404)(199,403)(200,402)
(201,397)(202,400)(203,399)(204,398)(205,429)(206,432)(207,431)(208,430)
(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)(216,422);;
s2 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 26)( 14, 25)
( 15, 28)( 16, 27)( 17, 30)( 18, 29)( 19, 32)( 20, 31)( 21, 34)( 22, 33)
( 23, 36)( 24, 35)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 62)( 50, 61)( 51, 64)( 52, 63)( 53, 66)( 54, 65)( 55, 68)( 56, 67)
( 57, 70)( 58, 69)( 59, 72)( 60, 71)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 98)( 86, 97)( 87,100)( 88, 99)( 89,102)( 90,101)
( 91,104)( 92,103)( 93,106)( 94,105)( 95,108)( 96,107)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,134)(122,133)(123,136)(124,135)
(125,138)(126,137)(127,140)(128,139)(129,142)(130,141)(131,144)(132,143)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,170)(158,169)
(159,172)(160,171)(161,174)(162,173)(163,176)(164,175)(165,178)(166,177)
(167,180)(168,179)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,206)(194,205)(195,208)(196,207)(197,210)(198,209)(199,212)(200,211)
(201,214)(202,213)(203,216)(204,215)(217,326)(218,325)(219,328)(220,327)
(221,330)(222,329)(223,332)(224,331)(225,334)(226,333)(227,336)(228,335)
(229,350)(230,349)(231,352)(232,351)(233,354)(234,353)(235,356)(236,355)
(237,358)(238,357)(239,360)(240,359)(241,338)(242,337)(243,340)(244,339)
(245,342)(246,341)(247,344)(248,343)(249,346)(250,345)(251,348)(252,347)
(253,362)(254,361)(255,364)(256,363)(257,366)(258,365)(259,368)(260,367)
(261,370)(262,369)(263,372)(264,371)(265,386)(266,385)(267,388)(268,387)
(269,390)(270,389)(271,392)(272,391)(273,394)(274,393)(275,396)(276,395)
(277,374)(278,373)(279,376)(280,375)(281,378)(282,377)(283,380)(284,379)
(285,382)(286,381)(287,384)(288,383)(289,398)(290,397)(291,400)(292,399)
(293,402)(294,401)(295,404)(296,403)(297,406)(298,405)(299,408)(300,407)
(301,422)(302,421)(303,424)(304,423)(305,426)(306,425)(307,428)(308,427)
(309,430)(310,429)(311,432)(312,431)(313,410)(314,409)(315,412)(316,411)
(317,414)(318,413)(319,416)(320,415)(321,418)(322,417)(323,420)(324,419);;
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*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 25)( 14, 26)
( 15, 28)( 16, 27)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 29)( 22, 30)
( 23, 32)( 24, 31)( 37, 81)( 38, 82)( 39, 84)( 40, 83)( 41, 77)( 42, 78)
( 43, 80)( 44, 79)( 45, 73)( 46, 74)( 47, 76)( 48, 75)( 49,105)( 50,106)
( 51,108)( 52,107)( 53,101)( 54,102)( 55,104)( 56,103)( 57, 97)( 58, 98)
( 59,100)( 60, 99)( 61, 93)( 62, 94)( 63, 96)( 64, 95)( 65, 89)( 66, 90)
( 67, 92)( 68, 91)( 69, 85)( 70, 86)( 71, 88)( 72, 87)(111,112)(113,117)
(114,118)(115,120)(116,119)(121,133)(122,134)(123,136)(124,135)(125,141)
(126,142)(127,144)(128,143)(129,137)(130,138)(131,140)(132,139)(145,189)
(146,190)(147,192)(148,191)(149,185)(150,186)(151,188)(152,187)(153,181)
(154,182)(155,184)(156,183)(157,213)(158,214)(159,216)(160,215)(161,209)
(162,210)(163,212)(164,211)(165,205)(166,206)(167,208)(168,207)(169,201)
(170,202)(171,204)(172,203)(173,197)(174,198)(175,200)(176,199)(177,193)
(178,194)(179,196)(180,195)(219,220)(221,225)(222,226)(223,228)(224,227)
(229,241)(230,242)(231,244)(232,243)(233,249)(234,250)(235,252)(236,251)
(237,245)(238,246)(239,248)(240,247)(253,297)(254,298)(255,300)(256,299)
(257,293)(258,294)(259,296)(260,295)(261,289)(262,290)(263,292)(264,291)
(265,321)(266,322)(267,324)(268,323)(269,317)(270,318)(271,320)(272,319)
(273,313)(274,314)(275,316)(276,315)(277,309)(278,310)(279,312)(280,311)
(281,305)(282,306)(283,308)(284,307)(285,301)(286,302)(287,304)(288,303)
(327,328)(329,333)(330,334)(331,336)(332,335)(337,349)(338,350)(339,352)
(340,351)(341,357)(342,358)(343,360)(344,359)(345,353)(346,354)(347,356)
(348,355)(361,405)(362,406)(363,408)(364,407)(365,401)(366,402)(367,404)
(368,403)(369,397)(370,398)(371,400)(372,399)(373,429)(374,430)(375,432)
(376,431)(377,425)(378,426)(379,428)(380,427)(381,421)(382,422)(383,424)
(384,423)(385,417)(386,418)(387,420)(388,419)(389,413)(390,414)(391,416)
(392,415)(393,409)(394,410)(395,412)(396,411);
s1 := Sym(432)!( 1,265)( 2,268)( 3,267)( 4,266)( 5,273)( 6,276)( 7,275)
( 8,274)( 9,269)( 10,272)( 11,271)( 12,270)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)
( 24,258)( 25,277)( 26,280)( 27,279)( 28,278)( 29,285)( 30,288)( 31,287)
( 32,286)( 33,281)( 34,284)( 35,283)( 36,282)( 37,229)( 38,232)( 39,231)
( 40,230)( 41,237)( 42,240)( 43,239)( 44,238)( 45,233)( 46,236)( 47,235)
( 48,234)( 49,217)( 50,220)( 51,219)( 52,218)( 53,225)( 54,228)( 55,227)
( 56,226)( 57,221)( 58,224)( 59,223)( 60,222)( 61,241)( 62,244)( 63,243)
( 64,242)( 65,249)( 66,252)( 67,251)( 68,250)( 69,245)( 70,248)( 71,247)
( 72,246)( 73,309)( 74,312)( 75,311)( 76,310)( 77,305)( 78,308)( 79,307)
( 80,306)( 81,301)( 82,304)( 83,303)( 84,302)( 85,297)( 86,300)( 87,299)
( 88,298)( 89,293)( 90,296)( 91,295)( 92,294)( 93,289)( 94,292)( 95,291)
( 96,290)( 97,321)( 98,324)( 99,323)(100,322)(101,317)(102,320)(103,319)
(104,318)(105,313)(106,316)(107,315)(108,314)(109,373)(110,376)(111,375)
(112,374)(113,381)(114,384)(115,383)(116,382)(117,377)(118,380)(119,379)
(120,378)(121,361)(122,364)(123,363)(124,362)(125,369)(126,372)(127,371)
(128,370)(129,365)(130,368)(131,367)(132,366)(133,385)(134,388)(135,387)
(136,386)(137,393)(138,396)(139,395)(140,394)(141,389)(142,392)(143,391)
(144,390)(145,337)(146,340)(147,339)(148,338)(149,345)(150,348)(151,347)
(152,346)(153,341)(154,344)(155,343)(156,342)(157,325)(158,328)(159,327)
(160,326)(161,333)(162,336)(163,335)(164,334)(165,329)(166,332)(167,331)
(168,330)(169,349)(170,352)(171,351)(172,350)(173,357)(174,360)(175,359)
(176,358)(177,353)(178,356)(179,355)(180,354)(181,417)(182,420)(183,419)
(184,418)(185,413)(186,416)(187,415)(188,414)(189,409)(190,412)(191,411)
(192,410)(193,405)(194,408)(195,407)(196,406)(197,401)(198,404)(199,403)
(200,402)(201,397)(202,400)(203,399)(204,398)(205,429)(206,432)(207,431)
(208,430)(209,425)(210,428)(211,427)(212,426)(213,421)(214,424)(215,423)
(216,422);
s2 := Sym(432)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 26)
( 14, 25)( 15, 28)( 16, 27)( 17, 30)( 18, 29)( 19, 32)( 20, 31)( 21, 34)
( 22, 33)( 23, 36)( 24, 35)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 62)( 50, 61)( 51, 64)( 52, 63)( 53, 66)( 54, 65)( 55, 68)
( 56, 67)( 57, 70)( 58, 69)( 59, 72)( 60, 71)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 98)( 86, 97)( 87,100)( 88, 99)( 89,102)
( 90,101)( 91,104)( 92,103)( 93,106)( 94,105)( 95,108)( 96,107)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,134)(122,133)(123,136)
(124,135)(125,138)(126,137)(127,140)(128,139)(129,142)(130,141)(131,144)
(132,143)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,170)
(158,169)(159,172)(160,171)(161,174)(162,173)(163,176)(164,175)(165,178)
(166,177)(167,180)(168,179)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,206)(194,205)(195,208)(196,207)(197,210)(198,209)(199,212)
(200,211)(201,214)(202,213)(203,216)(204,215)(217,326)(218,325)(219,328)
(220,327)(221,330)(222,329)(223,332)(224,331)(225,334)(226,333)(227,336)
(228,335)(229,350)(230,349)(231,352)(232,351)(233,354)(234,353)(235,356)
(236,355)(237,358)(238,357)(239,360)(240,359)(241,338)(242,337)(243,340)
(244,339)(245,342)(246,341)(247,344)(248,343)(249,346)(250,345)(251,348)
(252,347)(253,362)(254,361)(255,364)(256,363)(257,366)(258,365)(259,368)
(260,367)(261,370)(262,369)(263,372)(264,371)(265,386)(266,385)(267,388)
(268,387)(269,390)(270,389)(271,392)(272,391)(273,394)(274,393)(275,396)
(276,395)(277,374)(278,373)(279,376)(280,375)(281,378)(282,377)(283,380)
(284,379)(285,382)(286,381)(287,384)(288,383)(289,398)(290,397)(291,400)
(292,399)(293,402)(294,401)(295,404)(296,403)(297,406)(298,405)(299,408)
(300,407)(301,422)(302,421)(303,424)(304,423)(305,426)(306,425)(307,428)
(308,427)(309,430)(310,429)(311,432)(312,431)(313,410)(314,409)(315,412)
(316,411)(317,414)(318,413)(319,416)(320,415)(321,418)(322,417)(323,420)
(324,419);
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, s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope