Polytope of Type {18,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30216)
Rank : 3
Schlafli Type : {18,12}
Number of vertices, edges, etc : 72, 432, 48
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
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 : {6,12}*576e
4-fold quotients : {18,12}*432b, {9,6}*432
6-fold quotients : {6,6}*288b
8-fold quotients : {18,6}*216b
9-fold quotients : {6,12}*192a
12-fold quotients : {18,4}*144a, {6,12}*144c, {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 : {6,4}*48a, {3,6}*48, {6,3}*48
48-fold quotients : {9,2}*36, {3,6}*36
72-fold quotients : {3,3}*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, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)( 43, 84)
( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 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)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,181)(146,182)
(147,184)(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)(154,186)
(155,188)(156,187)(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)(229,249)
(230,250)(231,252)(232,251)(233,245)(234,246)(235,248)(236,247)(237,241)
(238,242)(239,244)(240,243)(253,289)(254,290)(255,292)(256,291)(257,297)
(258,298)(259,300)(260,299)(261,293)(262,294)(263,296)(264,295)(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)(337,357)(338,358)(339,360)(340,359)
(341,353)(342,354)(343,356)(344,355)(345,349)(346,350)(347,352)(348,351)
(361,397)(362,398)(363,400)(364,399)(365,405)(366,406)(367,408)(368,407)
(369,401)(370,402)(371,404)(372,403)(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)
( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)( 32,282)
( 33,277)( 34,280)( 35,279)( 36,278)( 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)( 61,249)( 62,252)( 63,251)( 64,250)
( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)( 72,242)
( 73,301)( 74,304)( 75,303)( 76,302)( 77,309)( 78,312)( 79,311)( 80,310)
( 81,305)( 82,308)( 83,307)( 84,306)( 85,289)( 86,292)( 87,291)( 88,290)
( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)( 96,294)
( 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)(133,393)(134,396)(135,395)(136,394)
(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)(144,386)
(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)
(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)
(177,349)(178,352)(179,351)(180,350)(181,409)(182,412)(183,411)(184,410)
(185,417)(186,420)(187,419)(188,418)(189,413)(190,416)(191,415)(192,414)
(193,397)(194,400)(195,399)(196,398)(197,405)(198,408)(199,407)(200,406)
(201,401)(202,404)(203,403)(204,402)(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)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)( 29, 30)
( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)( 43, 79)
( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)( 51, 87)
( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)( 59, 95)
( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)
( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)(117,118)
(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)(146,181)
(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)(154,189)
(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197)
(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)(170,205)
(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)(178,213)
(179,215)(180,216)(217,326)(218,325)(219,327)(220,328)(221,330)(222,329)
(223,331)(224,332)(225,334)(226,333)(227,335)(228,336)(229,338)(230,337)
(231,339)(232,340)(233,342)(234,341)(235,343)(236,344)(237,346)(238,345)
(239,347)(240,348)(241,350)(242,349)(243,351)(244,352)(245,354)(246,353)
(247,355)(248,356)(249,358)(250,357)(251,359)(252,360)(253,398)(254,397)
(255,399)(256,400)(257,402)(258,401)(259,403)(260,404)(261,406)(262,405)
(263,407)(264,408)(265,410)(266,409)(267,411)(268,412)(269,414)(270,413)
(271,415)(272,416)(273,418)(274,417)(275,419)(276,420)(277,422)(278,421)
(279,423)(280,424)(281,426)(282,425)(283,427)(284,428)(285,430)(286,429)
(287,431)(288,432)(289,362)(290,361)(291,363)(292,364)(293,366)(294,365)
(295,367)(296,368)(297,370)(298,369)(299,371)(300,372)(301,374)(302,373)
(303,375)(304,376)(305,378)(306,377)(307,379)(308,380)(309,382)(310,381)
(311,383)(312,384)(313,386)(314,385)(315,387)(316,388)(317,390)(318,389)
(319,391)(320,392)(321,394)(322,393)(323,395)(324,396);;
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*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 81)( 42, 82)
( 43, 84)( 44, 83)( 45, 77)( 46, 78)( 47, 80)( 48, 79)( 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)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,181)
(146,182)(147,184)(148,183)(149,189)(150,190)(151,192)(152,191)(153,185)
(154,186)(155,188)(156,187)(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)
(229,249)(230,250)(231,252)(232,251)(233,245)(234,246)(235,248)(236,247)
(237,241)(238,242)(239,244)(240,243)(253,289)(254,290)(255,292)(256,291)
(257,297)(258,298)(259,300)(260,299)(261,293)(262,294)(263,296)(264,295)
(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)(337,357)(338,358)(339,360)
(340,359)(341,353)(342,354)(343,356)(344,355)(345,349)(346,350)(347,352)
(348,351)(361,397)(362,398)(363,400)(364,399)(365,405)(366,406)(367,408)
(368,407)(369,401)(370,402)(371,404)(372,403)(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)( 25,285)( 26,288)( 27,287)( 28,286)( 29,281)( 30,284)( 31,283)
( 32,282)( 33,277)( 34,280)( 35,279)( 36,278)( 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)( 61,249)( 62,252)( 63,251)
( 64,250)( 65,245)( 66,248)( 67,247)( 68,246)( 69,241)( 70,244)( 71,243)
( 72,242)( 73,301)( 74,304)( 75,303)( 76,302)( 77,309)( 78,312)( 79,311)
( 80,310)( 81,305)( 82,308)( 83,307)( 84,306)( 85,289)( 86,292)( 87,291)
( 88,290)( 89,297)( 90,300)( 91,299)( 92,298)( 93,293)( 94,296)( 95,295)
( 96,294)( 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)(133,393)(134,396)(135,395)
(136,394)(137,389)(138,392)(139,391)(140,390)(141,385)(142,388)(143,387)
(144,386)(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)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)
(176,354)(177,349)(178,352)(179,351)(180,350)(181,409)(182,412)(183,411)
(184,410)(185,417)(186,420)(187,419)(188,418)(189,413)(190,416)(191,415)
(192,414)(193,397)(194,400)(195,399)(196,398)(197,405)(198,408)(199,407)
(200,406)(201,401)(202,404)(203,403)(204,402)(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)( 5, 6)( 9, 10)( 13, 14)( 17, 18)( 21, 22)( 25, 26)
( 29, 30)( 33, 34)( 37, 74)( 38, 73)( 39, 75)( 40, 76)( 41, 78)( 42, 77)
( 43, 79)( 44, 80)( 45, 82)( 46, 81)( 47, 83)( 48, 84)( 49, 86)( 50, 85)
( 51, 87)( 52, 88)( 53, 90)( 54, 89)( 55, 91)( 56, 92)( 57, 94)( 58, 93)
( 59, 95)( 60, 96)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)
( 67,103)( 68,104)( 69,106)( 70,105)( 71,107)( 72,108)(109,110)(113,114)
(117,118)(121,122)(125,126)(129,130)(133,134)(137,138)(141,142)(145,182)
(146,181)(147,183)(148,184)(149,186)(150,185)(151,187)(152,188)(153,190)
(154,189)(155,191)(156,192)(157,194)(158,193)(159,195)(160,196)(161,198)
(162,197)(163,199)(164,200)(165,202)(166,201)(167,203)(168,204)(169,206)
(170,205)(171,207)(172,208)(173,210)(174,209)(175,211)(176,212)(177,214)
(178,213)(179,215)(180,216)(217,326)(218,325)(219,327)(220,328)(221,330)
(222,329)(223,331)(224,332)(225,334)(226,333)(227,335)(228,336)(229,338)
(230,337)(231,339)(232,340)(233,342)(234,341)(235,343)(236,344)(237,346)
(238,345)(239,347)(240,348)(241,350)(242,349)(243,351)(244,352)(245,354)
(246,353)(247,355)(248,356)(249,358)(250,357)(251,359)(252,360)(253,398)
(254,397)(255,399)(256,400)(257,402)(258,401)(259,403)(260,404)(261,406)
(262,405)(263,407)(264,408)(265,410)(266,409)(267,411)(268,412)(269,414)
(270,413)(271,415)(272,416)(273,418)(274,417)(275,419)(276,420)(277,422)
(278,421)(279,423)(280,424)(281,426)(282,425)(283,427)(284,428)(285,430)
(286,429)(287,431)(288,432)(289,362)(290,361)(291,363)(292,364)(293,366)
(294,365)(295,367)(296,368)(297,370)(298,369)(299,371)(300,372)(301,374)
(302,373)(303,375)(304,376)(305,378)(306,377)(307,379)(308,380)(309,382)
(310,381)(311,383)(312,384)(313,386)(314,385)(315,387)(316,388)(317,390)
(318,389)(319,391)(320,392)(321,394)(322,393)(323,395)(324,396);
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*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*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