Polytope of Type {12,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1728e
if this polytope has a name.
Group : SmallGroup(1728,30326)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 144, 432, 72
Order of s0s1s2 : 12
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 : {6,6}*864b
3-fold quotients : {12,6}*576e
4-fold quotients : {12,6}*432c, {6,3}*432
6-fold quotients : {6,6}*288a
8-fold quotients : {6,6}*216a
9-fold quotients : {12,6}*192a
12-fold quotients : {12,6}*144c, {6,3}*144
16-fold quotients : {6,3}*108
18-fold quotients : {6,6}*96
24-fold quotients : {6,6}*72b
36-fold quotients : {4,6}*48a, {3,6}*48, {6,3}*48
48-fold quotients : {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)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 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, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 93)
( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,105)( 66,106)( 67,108)( 68,107)( 69,101)
( 70,102)( 71,104)( 72,103)(111,112)(113,117)(114,118)(115,120)(116,119)
(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)(138,142)
(139,144)(140,143)(145,181)(146,182)(147,184)(148,183)(149,189)(150,190)
(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,193)(158,194)
(159,196)(160,195)(161,201)(162,202)(163,204)(164,203)(165,197)(166,198)
(167,200)(168,199)(169,205)(170,206)(171,208)(172,207)(173,213)(174,214)
(175,216)(176,215)(177,209)(178,210)(179,212)(180,211)(217,325)(218,326)
(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)(226,330)
(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)(234,346)
(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)(242,350)
(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)(250,354)
(251,356)(252,355)(253,397)(254,398)(255,400)(256,399)(257,405)(258,406)
(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,409)(266,410)
(267,412)(268,411)(269,417)(270,418)(271,420)(272,419)(273,413)(274,414)
(275,416)(276,415)(277,421)(278,422)(279,424)(280,423)(281,429)(282,430)
(283,432)(284,431)(285,425)(286,426)(287,428)(288,427)(289,361)(290,362)
(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)(298,366)
(299,368)(300,367)(301,373)(302,374)(303,376)(304,375)(305,381)(306,382)
(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,385)(314,386)
(315,388)(316,387)(317,393)(318,394)(319,396)(320,395)(321,389)(322,390)
(323,392)(324,391);;
s1 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)( 8,258)
( 9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)( 16,286)
( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)( 48,226)
( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)( 56,242)
( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)( 64,234)
( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)( 96,318)
( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)(104,310)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)(120,370)
(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)(128,386)
(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)(136,378)
(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)(144,374)
(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)
(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)(160,358)
(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)(168,354)
(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)(176,346)
(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)
(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)(192,406)
(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)(200,422)
(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)(208,414)
(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)(216,410);;
s2 := ( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 18)( 6, 17)( 7, 19)( 8, 20)
( 9, 22)( 10, 21)( 11, 23)( 12, 24)( 25, 26)( 29, 30)( 33, 34)( 37, 86)
( 38, 85)( 39, 87)( 40, 88)( 41, 90)( 42, 89)( 43, 91)( 44, 92)( 45, 94)
( 46, 93)( 47, 95)( 48, 96)( 49, 74)( 50, 73)( 51, 75)( 52, 76)( 53, 78)
( 54, 77)( 55, 79)( 56, 80)( 57, 82)( 58, 81)( 59, 83)( 60, 84)( 61, 98)
( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)( 68,104)( 69,106)
( 70,105)( 71,107)( 72,108)(109,122)(110,121)(111,123)(112,124)(113,126)
(114,125)(115,127)(116,128)(117,130)(118,129)(119,131)(120,132)(133,134)
(137,138)(141,142)(145,194)(146,193)(147,195)(148,196)(149,198)(150,197)
(151,199)(152,200)(153,202)(154,201)(155,203)(156,204)(157,182)(158,181)
(159,183)(160,184)(161,186)(162,185)(163,187)(164,188)(165,190)(166,189)
(167,191)(168,192)(169,206)(170,205)(171,207)(172,208)(173,210)(174,209)
(175,211)(176,212)(177,214)(178,213)(179,215)(180,216)(217,230)(218,229)
(219,231)(220,232)(221,234)(222,233)(223,235)(224,236)(225,238)(226,237)
(227,239)(228,240)(241,242)(245,246)(249,250)(253,302)(254,301)(255,303)
(256,304)(257,306)(258,305)(259,307)(260,308)(261,310)(262,309)(263,311)
(264,312)(265,290)(266,289)(267,291)(268,292)(269,294)(270,293)(271,295)
(272,296)(273,298)(274,297)(275,299)(276,300)(277,314)(278,313)(279,315)
(280,316)(281,318)(282,317)(283,319)(284,320)(285,322)(286,321)(287,323)
(288,324)(325,338)(326,337)(327,339)(328,340)(329,342)(330,341)(331,343)
(332,344)(333,346)(334,345)(335,347)(336,348)(349,350)(353,354)(357,358)
(361,410)(362,409)(363,411)(364,412)(365,414)(366,413)(367,415)(368,416)
(369,418)(370,417)(371,419)(372,420)(373,398)(374,397)(375,399)(376,400)
(377,402)(378,401)(379,403)(380,404)(381,406)(382,405)(383,407)(384,408)
(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)(392,428)
(393,430)(394,429)(395,431)(396,432);;
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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 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, 85)( 50, 86)( 51, 88)( 52, 87)
( 53, 93)( 54, 94)( 55, 96)( 56, 95)( 57, 89)( 58, 90)( 59, 92)( 60, 91)
( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,105)( 66,106)( 67,108)( 68,107)
( 69,101)( 70,102)( 71,104)( 72,103)(111,112)(113,117)(114,118)(115,120)
(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)(137,141)
(138,142)(139,144)(140,143)(145,181)(146,182)(147,184)(148,183)(149,189)
(150,190)(151,192)(152,191)(153,185)(154,186)(155,188)(156,187)(157,193)
(158,194)(159,196)(160,195)(161,201)(162,202)(163,204)(164,203)(165,197)
(166,198)(167,200)(168,199)(169,205)(170,206)(171,208)(172,207)(173,213)
(174,214)(175,216)(176,215)(177,209)(178,210)(179,212)(180,211)(217,325)
(218,326)(219,328)(220,327)(221,333)(222,334)(223,336)(224,335)(225,329)
(226,330)(227,332)(228,331)(229,337)(230,338)(231,340)(232,339)(233,345)
(234,346)(235,348)(236,347)(237,341)(238,342)(239,344)(240,343)(241,349)
(242,350)(243,352)(244,351)(245,357)(246,358)(247,360)(248,359)(249,353)
(250,354)(251,356)(252,355)(253,397)(254,398)(255,400)(256,399)(257,405)
(258,406)(259,408)(260,407)(261,401)(262,402)(263,404)(264,403)(265,409)
(266,410)(267,412)(268,411)(269,417)(270,418)(271,420)(272,419)(273,413)
(274,414)(275,416)(276,415)(277,421)(278,422)(279,424)(280,423)(281,429)
(282,430)(283,432)(284,431)(285,425)(286,426)(287,428)(288,427)(289,361)
(290,362)(291,364)(292,363)(293,369)(294,370)(295,372)(296,371)(297,365)
(298,366)(299,368)(300,367)(301,373)(302,374)(303,376)(304,375)(305,381)
(306,382)(307,384)(308,383)(309,377)(310,378)(311,380)(312,379)(313,385)
(314,386)(315,388)(316,387)(317,393)(318,394)(319,396)(320,395)(321,389)
(322,390)(323,392)(324,391);
s1 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)
( 8,258)( 9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)
( 16,286)( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)
( 48,226)( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)
( 56,242)( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)
( 64,234)( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)
( 96,318)( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)
(104,310)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)
(120,370)(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)
(128,386)(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)
(136,378)(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)
(144,374)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)
(152,330)(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)
(160,358)(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)
(168,354)(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)
(176,346)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)
(184,398)(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)
(192,406)(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)
(200,422)(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)
(208,414)(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)
(216,410);
s2 := Sym(432)!( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 18)( 6, 17)( 7, 19)
( 8, 20)( 9, 22)( 10, 21)( 11, 23)( 12, 24)( 25, 26)( 29, 30)( 33, 34)
( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 90)( 42, 89)( 43, 91)( 44, 92)
( 45, 94)( 46, 93)( 47, 95)( 48, 96)( 49, 74)( 50, 73)( 51, 75)( 52, 76)
( 53, 78)( 54, 77)( 55, 79)( 56, 80)( 57, 82)( 58, 81)( 59, 83)( 60, 84)
( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,102)( 66,101)( 67,103)( 68,104)
( 69,106)( 70,105)( 71,107)( 72,108)(109,122)(110,121)(111,123)(112,124)
(113,126)(114,125)(115,127)(116,128)(117,130)(118,129)(119,131)(120,132)
(133,134)(137,138)(141,142)(145,194)(146,193)(147,195)(148,196)(149,198)
(150,197)(151,199)(152,200)(153,202)(154,201)(155,203)(156,204)(157,182)
(158,181)(159,183)(160,184)(161,186)(162,185)(163,187)(164,188)(165,190)
(166,189)(167,191)(168,192)(169,206)(170,205)(171,207)(172,208)(173,210)
(174,209)(175,211)(176,212)(177,214)(178,213)(179,215)(180,216)(217,230)
(218,229)(219,231)(220,232)(221,234)(222,233)(223,235)(224,236)(225,238)
(226,237)(227,239)(228,240)(241,242)(245,246)(249,250)(253,302)(254,301)
(255,303)(256,304)(257,306)(258,305)(259,307)(260,308)(261,310)(262,309)
(263,311)(264,312)(265,290)(266,289)(267,291)(268,292)(269,294)(270,293)
(271,295)(272,296)(273,298)(274,297)(275,299)(276,300)(277,314)(278,313)
(279,315)(280,316)(281,318)(282,317)(283,319)(284,320)(285,322)(286,321)
(287,323)(288,324)(325,338)(326,337)(327,339)(328,340)(329,342)(330,341)
(331,343)(332,344)(333,346)(334,345)(335,347)(336,348)(349,350)(353,354)
(357,358)(361,410)(362,409)(363,411)(364,412)(365,414)(366,413)(367,415)
(368,416)(369,418)(370,417)(371,419)(372,420)(373,398)(374,397)(375,399)
(376,400)(377,402)(378,401)(379,403)(380,404)(381,406)(382,405)(383,407)
(384,408)(385,422)(386,421)(387,423)(388,424)(389,426)(390,425)(391,427)
(392,428)(393,430)(394,429)(395,431)(396,432);
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*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s2*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope