Polytope of Type {12,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,18}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30216)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 48, 432, 72
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,18}*864
3-fold quotients : {12,6}*576e
4-fold quotients : {12,18}*432b, {6,9}*432
6-fold quotients : {6,6}*288a
8-fold quotients : {6,18}*216b
9-fold quotients : {12,6}*192a
12-fold quotients : {4,18}*144a, {12,6}*144c, {6,3}*144
16-fold quotients : {6,9}*108
18-fold quotients : {6,6}*96
24-fold quotients : {2,18}*72, {6,6}*72b
36-fold quotients : {4,6}*48a, {3,6}*48, {6,3}*48
48-fold quotients : {2,9}*36, {6,3}*36
72-fold quotients : {3,3}*24, {2,6}*24
108-fold quotients : {4,2}*16
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)
( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)
( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)
( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)
( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)
(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)
(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)
(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)
(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)
(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)
(179,216)(180,215)(217,325)(218,326)(219,328)(220,327)(221,329)(222,330)
(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,337)(230,338)
(231,340)(232,339)(233,341)(234,342)(235,344)(236,343)(237,345)(238,346)
(239,348)(240,347)(241,349)(242,350)(243,352)(244,351)(245,353)(246,354)
(247,356)(248,355)(249,357)(250,358)(251,360)(252,359)(253,397)(254,398)
(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)(262,406)
(263,408)(264,407)(265,409)(266,410)(267,412)(268,411)(269,413)(270,414)
(271,416)(272,415)(273,417)(274,418)(275,420)(276,419)(277,421)(278,422)
(279,424)(280,423)(281,425)(282,426)(283,428)(284,427)(285,429)(286,430)
(287,432)(288,431)(289,361)(290,362)(291,364)(292,363)(293,365)(294,366)
(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,373)(302,374)
(303,376)(304,375)(305,377)(306,378)(307,380)(308,379)(309,381)(310,382)
(311,384)(312,383)(313,385)(314,386)(315,388)(316,387)(317,389)(318,390)
(319,392)(320,391)(321,393)(322,394)(323,396)(324,395);;
s1 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)( 8,262)
( 9,257)( 10,260)( 11,259)( 12,258)( 13,285)( 14,288)( 15,287)( 16,286)
( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)( 24,278)
( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)( 32,270)
( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)( 48,222)
( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)( 56,246)
( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)( 64,238)
( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)( 80,298)
( 81,293)( 82,296)( 83,295)( 84,294)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)( 96,314)
( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)(104,306)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)(120,366)
(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)(128,390)
(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)(136,382)
(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)(144,374)
(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)(152,334)
(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)(160,358)
(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)(168,350)
(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)(176,342)
(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)
(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)(192,402)
(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)(200,426)
(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)(208,418)
(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)(216,410);;
s2 := ( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 22)( 6, 21)( 7, 23)( 8, 24)
( 9, 18)( 10, 17)( 11, 19)( 12, 20)( 25, 34)( 26, 33)( 27, 35)( 28, 36)
( 29, 30)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 94)( 42, 93)( 43, 95)
( 44, 96)( 45, 90)( 46, 89)( 47, 91)( 48, 92)( 49, 74)( 50, 73)( 51, 75)
( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)( 58, 77)( 59, 79)
( 60, 80)( 61,106)( 62,105)( 63,107)( 64,108)( 65,102)( 66,101)( 67,103)
( 68,104)( 69, 98)( 70, 97)( 71, 99)( 72,100)(109,122)(110,121)(111,123)
(112,124)(113,130)(114,129)(115,131)(116,132)(117,126)(118,125)(119,127)
(120,128)(133,142)(134,141)(135,143)(136,144)(137,138)(145,194)(146,193)
(147,195)(148,196)(149,202)(150,201)(151,203)(152,204)(153,198)(154,197)
(155,199)(156,200)(157,182)(158,181)(159,183)(160,184)(161,190)(162,189)
(163,191)(164,192)(165,186)(166,185)(167,187)(168,188)(169,214)(170,213)
(171,215)(172,216)(173,210)(174,209)(175,211)(176,212)(177,206)(178,205)
(179,207)(180,208)(217,230)(218,229)(219,231)(220,232)(221,238)(222,237)
(223,239)(224,240)(225,234)(226,233)(227,235)(228,236)(241,250)(242,249)
(243,251)(244,252)(245,246)(253,302)(254,301)(255,303)(256,304)(257,310)
(258,309)(259,311)(260,312)(261,306)(262,305)(263,307)(264,308)(265,290)
(266,289)(267,291)(268,292)(269,298)(270,297)(271,299)(272,300)(273,294)
(274,293)(275,295)(276,296)(277,322)(278,321)(279,323)(280,324)(281,318)
(282,317)(283,319)(284,320)(285,314)(286,313)(287,315)(288,316)(325,338)
(326,337)(327,339)(328,340)(329,346)(330,345)(331,347)(332,348)(333,342)
(334,341)(335,343)(336,344)(349,358)(350,357)(351,359)(352,360)(353,354)
(361,410)(362,409)(363,411)(364,412)(365,418)(366,417)(367,419)(368,420)
(369,414)(370,413)(371,415)(372,416)(373,398)(374,397)(375,399)(376,400)
(377,406)(378,405)(379,407)(380,408)(381,402)(382,401)(383,403)(384,404)
(385,430)(386,429)(387,431)(388,432)(389,426)(390,425)(391,427)(392,428)
(393,422)(394,421)(395,423)(396,424);;
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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*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,
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(432)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)
( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)
( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)
( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)
( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)
(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)
(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)
(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)
(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)
(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)
(178,214)(179,216)(180,215)(217,325)(218,326)(219,328)(220,327)(221,329)
(222,330)(223,332)(224,331)(225,333)(226,334)(227,336)(228,335)(229,337)
(230,338)(231,340)(232,339)(233,341)(234,342)(235,344)(236,343)(237,345)
(238,346)(239,348)(240,347)(241,349)(242,350)(243,352)(244,351)(245,353)
(246,354)(247,356)(248,355)(249,357)(250,358)(251,360)(252,359)(253,397)
(254,398)(255,400)(256,399)(257,401)(258,402)(259,404)(260,403)(261,405)
(262,406)(263,408)(264,407)(265,409)(266,410)(267,412)(268,411)(269,413)
(270,414)(271,416)(272,415)(273,417)(274,418)(275,420)(276,419)(277,421)
(278,422)(279,424)(280,423)(281,425)(282,426)(283,428)(284,427)(285,429)
(286,430)(287,432)(288,431)(289,361)(290,362)(291,364)(292,363)(293,365)
(294,366)(295,368)(296,367)(297,369)(298,370)(299,372)(300,371)(301,373)
(302,374)(303,376)(304,375)(305,377)(306,378)(307,380)(308,379)(309,381)
(310,382)(311,384)(312,383)(313,385)(314,386)(315,388)(316,387)(317,389)
(318,390)(319,392)(320,391)(321,393)(322,394)(323,396)(324,395);
s1 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,261)( 6,264)( 7,263)
( 8,262)( 9,257)( 10,260)( 11,259)( 12,258)( 13,285)( 14,288)( 15,287)
( 16,286)( 17,281)( 18,284)( 19,283)( 20,282)( 21,277)( 22,280)( 23,279)
( 24,278)( 25,273)( 26,276)( 27,275)( 28,274)( 29,269)( 30,272)( 31,271)
( 32,270)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,225)( 42,228)( 43,227)( 44,226)( 45,221)( 46,224)( 47,223)
( 48,222)( 49,249)( 50,252)( 51,251)( 52,250)( 53,245)( 54,248)( 55,247)
( 56,246)( 57,241)( 58,244)( 59,243)( 60,242)( 61,237)( 62,240)( 63,239)
( 64,238)( 65,233)( 66,236)( 67,235)( 68,234)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,297)( 78,300)( 79,299)
( 80,298)( 81,293)( 82,296)( 83,295)( 84,294)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,317)( 90,320)( 91,319)( 92,318)( 93,313)( 94,316)( 95,315)
( 96,314)( 97,309)( 98,312)( 99,311)(100,310)(101,305)(102,308)(103,307)
(104,306)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,369)(114,372)(115,371)(116,370)(117,365)(118,368)(119,367)
(120,366)(121,393)(122,396)(123,395)(124,394)(125,389)(126,392)(127,391)
(128,390)(129,385)(130,388)(131,387)(132,386)(133,381)(134,384)(135,383)
(136,382)(137,377)(138,380)(139,379)(140,378)(141,373)(142,376)(143,375)
(144,374)(145,325)(146,328)(147,327)(148,326)(149,333)(150,336)(151,335)
(152,334)(153,329)(154,332)(155,331)(156,330)(157,357)(158,360)(159,359)
(160,358)(161,353)(162,356)(163,355)(164,354)(165,349)(166,352)(167,351)
(168,350)(169,345)(170,348)(171,347)(172,346)(173,341)(174,344)(175,343)
(176,342)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)
(184,398)(185,405)(186,408)(187,407)(188,406)(189,401)(190,404)(191,403)
(192,402)(193,429)(194,432)(195,431)(196,430)(197,425)(198,428)(199,427)
(200,426)(201,421)(202,424)(203,423)(204,422)(205,417)(206,420)(207,419)
(208,418)(209,413)(210,416)(211,415)(212,414)(213,409)(214,412)(215,411)
(216,410);
s2 := Sym(432)!( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 22)( 6, 21)( 7, 23)
( 8, 24)( 9, 18)( 10, 17)( 11, 19)( 12, 20)( 25, 34)( 26, 33)( 27, 35)
( 28, 36)( 29, 30)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 94)( 42, 93)
( 43, 95)( 44, 96)( 45, 90)( 46, 89)( 47, 91)( 48, 92)( 49, 74)( 50, 73)
( 51, 75)( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)( 58, 77)
( 59, 79)( 60, 80)( 61,106)( 62,105)( 63,107)( 64,108)( 65,102)( 66,101)
( 67,103)( 68,104)( 69, 98)( 70, 97)( 71, 99)( 72,100)(109,122)(110,121)
(111,123)(112,124)(113,130)(114,129)(115,131)(116,132)(117,126)(118,125)
(119,127)(120,128)(133,142)(134,141)(135,143)(136,144)(137,138)(145,194)
(146,193)(147,195)(148,196)(149,202)(150,201)(151,203)(152,204)(153,198)
(154,197)(155,199)(156,200)(157,182)(158,181)(159,183)(160,184)(161,190)
(162,189)(163,191)(164,192)(165,186)(166,185)(167,187)(168,188)(169,214)
(170,213)(171,215)(172,216)(173,210)(174,209)(175,211)(176,212)(177,206)
(178,205)(179,207)(180,208)(217,230)(218,229)(219,231)(220,232)(221,238)
(222,237)(223,239)(224,240)(225,234)(226,233)(227,235)(228,236)(241,250)
(242,249)(243,251)(244,252)(245,246)(253,302)(254,301)(255,303)(256,304)
(257,310)(258,309)(259,311)(260,312)(261,306)(262,305)(263,307)(264,308)
(265,290)(266,289)(267,291)(268,292)(269,298)(270,297)(271,299)(272,300)
(273,294)(274,293)(275,295)(276,296)(277,322)(278,321)(279,323)(280,324)
(281,318)(282,317)(283,319)(284,320)(285,314)(286,313)(287,315)(288,316)
(325,338)(326,337)(327,339)(328,340)(329,346)(330,345)(331,347)(332,348)
(333,342)(334,341)(335,343)(336,344)(349,358)(350,357)(351,359)(352,360)
(353,354)(361,410)(362,409)(363,411)(364,412)(365,418)(366,417)(367,419)
(368,420)(369,414)(370,413)(371,415)(372,416)(373,398)(374,397)(375,399)
(376,400)(377,406)(378,405)(379,407)(380,408)(381,402)(382,401)(383,403)
(384,404)(385,430)(386,429)(387,431)(388,432)(389,426)(390,425)(391,427)
(392,428)(393,422)(394,421)(395,423)(396,424);
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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*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,
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