Polytope of Type {12,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3538)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 12
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 : {12,12}*864a
3-fold quotients : {12,12}*576c
4-fold quotients : {12,6}*432a, {6,12}*432c
6-fold quotients : {12,12}*288c
8-fold quotients : {6,6}*216c
9-fold quotients : {12,4}*192a
12-fold quotients : {12,6}*144b, {6,12}*144c
16-fold quotients : {3,6}*108
18-fold quotients : {12,4}*96a
24-fold quotients : {6,6}*72c
27-fold quotients : {4,4}*64
36-fold quotients : {12,2}*48, {6,4}*48a
48-fold quotients : {3,6}*36
54-fold quotients : {4,4}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*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 := ( 4, 7)( 5, 8)( 6, 9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)
( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)
( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)
( 79, 94)( 80, 95)( 81, 96)(109,136)(110,137)(111,138)(112,142)(113,143)
(114,144)(115,139)(116,140)(117,141)(118,154)(119,155)(120,156)(121,160)
(122,161)(123,162)(124,157)(125,158)(126,159)(127,145)(128,146)(129,147)
(130,151)(131,152)(132,153)(133,148)(134,149)(135,150)(166,169)(167,170)
(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)(178,184)
(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)(201,210)
(202,214)(203,215)(204,216)(205,211)(206,212)(207,213)(217,406)(218,407)
(219,408)(220,412)(221,413)(222,414)(223,409)(224,410)(225,411)(226,424)
(227,425)(228,426)(229,430)(230,431)(231,432)(232,427)(233,428)(234,429)
(235,415)(236,416)(237,417)(238,421)(239,422)(240,423)(241,418)(242,419)
(243,420)(244,379)(245,380)(246,381)(247,385)(248,386)(249,387)(250,382)
(251,383)(252,384)(253,397)(254,398)(255,399)(256,403)(257,404)(258,405)
(259,400)(260,401)(261,402)(262,388)(263,389)(264,390)(265,394)(266,395)
(267,396)(268,391)(269,392)(270,393)(271,325)(272,326)(273,327)(274,331)
(275,332)(276,333)(277,328)(278,329)(279,330)(280,343)(281,344)(282,345)
(283,349)(284,350)(285,351)(286,346)(287,347)(288,348)(289,334)(290,335)
(291,336)(292,340)(293,341)(294,342)(295,337)(296,338)(297,339)(298,352)
(299,353)(300,354)(301,358)(302,359)(303,360)(304,355)(305,356)(306,357)
(307,370)(308,371)(309,372)(310,376)(311,377)(312,378)(313,373)(314,374)
(315,375)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)(322,364)
(323,365)(324,366);;
s1 := ( 1,239)( 2,240)( 3,238)( 4,237)( 5,235)( 6,236)( 7,241)( 8,242)
( 9,243)( 10,230)( 11,231)( 12,229)( 13,228)( 14,226)( 15,227)( 16,232)
( 17,233)( 18,234)( 19,221)( 20,222)( 21,220)( 22,219)( 23,217)( 24,218)
( 25,223)( 26,224)( 27,225)( 28,266)( 29,267)( 30,265)( 31,264)( 32,262)
( 33,263)( 34,268)( 35,269)( 36,270)( 37,257)( 38,258)( 39,256)( 40,255)
( 41,253)( 42,254)( 43,259)( 44,260)( 45,261)( 46,248)( 47,249)( 48,247)
( 49,246)( 50,244)( 51,245)( 52,250)( 53,251)( 54,252)( 55,293)( 56,294)
( 57,292)( 58,291)( 59,289)( 60,290)( 61,295)( 62,296)( 63,297)( 64,284)
( 65,285)( 66,283)( 67,282)( 68,280)( 69,281)( 70,286)( 71,287)( 72,288)
( 73,275)( 74,276)( 75,274)( 76,273)( 77,271)( 78,272)( 79,277)( 80,278)
( 81,279)( 82,320)( 83,321)( 84,319)( 85,318)( 86,316)( 87,317)( 88,322)
( 89,323)( 90,324)( 91,311)( 92,312)( 93,310)( 94,309)( 95,307)( 96,308)
( 97,313)( 98,314)( 99,315)(100,302)(101,303)(102,301)(103,300)(104,298)
(105,299)(106,304)(107,305)(108,306)(109,347)(110,348)(111,346)(112,345)
(113,343)(114,344)(115,349)(116,350)(117,351)(118,338)(119,339)(120,337)
(121,336)(122,334)(123,335)(124,340)(125,341)(126,342)(127,329)(128,330)
(129,328)(130,327)(131,325)(132,326)(133,331)(134,332)(135,333)(136,374)
(137,375)(138,373)(139,372)(140,370)(141,371)(142,376)(143,377)(144,378)
(145,365)(146,366)(147,364)(148,363)(149,361)(150,362)(151,367)(152,368)
(153,369)(154,356)(155,357)(156,355)(157,354)(158,352)(159,353)(160,358)
(161,359)(162,360)(163,401)(164,402)(165,400)(166,399)(167,397)(168,398)
(169,403)(170,404)(171,405)(172,392)(173,393)(174,391)(175,390)(176,388)
(177,389)(178,394)(179,395)(180,396)(181,383)(182,384)(183,382)(184,381)
(185,379)(186,380)(187,385)(188,386)(189,387)(190,428)(191,429)(192,427)
(193,426)(194,424)(195,425)(196,430)(197,431)(198,432)(199,419)(200,420)
(201,418)(202,417)(203,415)(204,416)(205,421)(206,422)(207,423)(208,410)
(209,411)(210,409)(211,408)(212,406)(213,407)(214,412)(215,413)(216,414);;
s2 := ( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)
(117,143)(118,154)(119,156)(120,155)(121,157)(122,159)(123,158)(124,160)
(125,162)(126,161)(127,145)(128,147)(129,146)(130,148)(131,150)(132,149)
(133,151)(134,153)(135,152)(163,190)(164,192)(165,191)(166,193)(167,195)
(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)(175,211)
(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)(183,200)
(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(217,271)(218,273)
(219,272)(220,274)(221,276)(222,275)(223,277)(224,279)(225,278)(226,289)
(227,291)(228,290)(229,292)(230,294)(231,293)(232,295)(233,297)(234,296)
(235,280)(236,282)(237,281)(238,283)(239,285)(240,284)(241,286)(242,288)
(243,287)(244,298)(245,300)(246,299)(247,301)(248,303)(249,302)(250,304)
(251,306)(252,305)(253,316)(254,318)(255,317)(256,319)(257,321)(258,320)
(259,322)(260,324)(261,323)(262,307)(263,309)(264,308)(265,310)(266,312)
(267,311)(268,313)(269,315)(270,314)(325,406)(326,408)(327,407)(328,409)
(329,411)(330,410)(331,412)(332,414)(333,413)(334,424)(335,426)(336,425)
(337,427)(338,429)(339,428)(340,430)(341,432)(342,431)(343,415)(344,417)
(345,416)(346,418)(347,420)(348,419)(349,421)(350,423)(351,422)(352,379)
(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)(359,387)(360,386)
(361,397)(362,399)(363,398)(364,400)(365,402)(366,401)(367,403)(368,405)
(369,404)(370,388)(371,390)(372,389)(373,391)(374,393)(375,392)(376,394)
(377,396)(378,395);;
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*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s1*s2*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 4, 7)( 5, 8)( 6, 9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)
( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)
( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)
( 78, 99)( 79, 94)( 80, 95)( 81, 96)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,154)(119,155)(120,156)
(121,160)(122,161)(123,162)(124,157)(125,158)(126,159)(127,145)(128,146)
(129,147)(130,151)(131,152)(132,153)(133,148)(134,149)(135,150)(166,169)
(167,170)(168,171)(172,181)(173,182)(174,183)(175,187)(176,188)(177,189)
(178,184)(179,185)(180,186)(193,196)(194,197)(195,198)(199,208)(200,209)
(201,210)(202,214)(203,215)(204,216)(205,211)(206,212)(207,213)(217,406)
(218,407)(219,408)(220,412)(221,413)(222,414)(223,409)(224,410)(225,411)
(226,424)(227,425)(228,426)(229,430)(230,431)(231,432)(232,427)(233,428)
(234,429)(235,415)(236,416)(237,417)(238,421)(239,422)(240,423)(241,418)
(242,419)(243,420)(244,379)(245,380)(246,381)(247,385)(248,386)(249,387)
(250,382)(251,383)(252,384)(253,397)(254,398)(255,399)(256,403)(257,404)
(258,405)(259,400)(260,401)(261,402)(262,388)(263,389)(264,390)(265,394)
(266,395)(267,396)(268,391)(269,392)(270,393)(271,325)(272,326)(273,327)
(274,331)(275,332)(276,333)(277,328)(278,329)(279,330)(280,343)(281,344)
(282,345)(283,349)(284,350)(285,351)(286,346)(287,347)(288,348)(289,334)
(290,335)(291,336)(292,340)(293,341)(294,342)(295,337)(296,338)(297,339)
(298,352)(299,353)(300,354)(301,358)(302,359)(303,360)(304,355)(305,356)
(306,357)(307,370)(308,371)(309,372)(310,376)(311,377)(312,378)(313,373)
(314,374)(315,375)(316,361)(317,362)(318,363)(319,367)(320,368)(321,369)
(322,364)(323,365)(324,366);
s1 := Sym(432)!( 1,239)( 2,240)( 3,238)( 4,237)( 5,235)( 6,236)( 7,241)
( 8,242)( 9,243)( 10,230)( 11,231)( 12,229)( 13,228)( 14,226)( 15,227)
( 16,232)( 17,233)( 18,234)( 19,221)( 20,222)( 21,220)( 22,219)( 23,217)
( 24,218)( 25,223)( 26,224)( 27,225)( 28,266)( 29,267)( 30,265)( 31,264)
( 32,262)( 33,263)( 34,268)( 35,269)( 36,270)( 37,257)( 38,258)( 39,256)
( 40,255)( 41,253)( 42,254)( 43,259)( 44,260)( 45,261)( 46,248)( 47,249)
( 48,247)( 49,246)( 50,244)( 51,245)( 52,250)( 53,251)( 54,252)( 55,293)
( 56,294)( 57,292)( 58,291)( 59,289)( 60,290)( 61,295)( 62,296)( 63,297)
( 64,284)( 65,285)( 66,283)( 67,282)( 68,280)( 69,281)( 70,286)( 71,287)
( 72,288)( 73,275)( 74,276)( 75,274)( 76,273)( 77,271)( 78,272)( 79,277)
( 80,278)( 81,279)( 82,320)( 83,321)( 84,319)( 85,318)( 86,316)( 87,317)
( 88,322)( 89,323)( 90,324)( 91,311)( 92,312)( 93,310)( 94,309)( 95,307)
( 96,308)( 97,313)( 98,314)( 99,315)(100,302)(101,303)(102,301)(103,300)
(104,298)(105,299)(106,304)(107,305)(108,306)(109,347)(110,348)(111,346)
(112,345)(113,343)(114,344)(115,349)(116,350)(117,351)(118,338)(119,339)
(120,337)(121,336)(122,334)(123,335)(124,340)(125,341)(126,342)(127,329)
(128,330)(129,328)(130,327)(131,325)(132,326)(133,331)(134,332)(135,333)
(136,374)(137,375)(138,373)(139,372)(140,370)(141,371)(142,376)(143,377)
(144,378)(145,365)(146,366)(147,364)(148,363)(149,361)(150,362)(151,367)
(152,368)(153,369)(154,356)(155,357)(156,355)(157,354)(158,352)(159,353)
(160,358)(161,359)(162,360)(163,401)(164,402)(165,400)(166,399)(167,397)
(168,398)(169,403)(170,404)(171,405)(172,392)(173,393)(174,391)(175,390)
(176,388)(177,389)(178,394)(179,395)(180,396)(181,383)(182,384)(183,382)
(184,381)(185,379)(186,380)(187,385)(188,386)(189,387)(190,428)(191,429)
(192,427)(193,426)(194,424)(195,425)(196,430)(197,431)(198,432)(199,419)
(200,420)(201,418)(202,417)(203,415)(204,416)(205,421)(206,422)(207,423)
(208,410)(209,411)(210,409)(211,408)(212,406)(213,407)(214,412)(215,413)
(216,414);
s2 := Sym(432)!( 2, 3)( 5, 6)( 8, 9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)
(116,144)(117,143)(118,154)(119,156)(120,155)(121,157)(122,159)(123,158)
(124,160)(125,162)(126,161)(127,145)(128,147)(129,146)(130,148)(131,150)
(132,149)(133,151)(134,153)(135,152)(163,190)(164,192)(165,191)(166,193)
(167,195)(168,194)(169,196)(170,198)(171,197)(172,208)(173,210)(174,209)
(175,211)(176,213)(177,212)(178,214)(179,216)(180,215)(181,199)(182,201)
(183,200)(184,202)(185,204)(186,203)(187,205)(188,207)(189,206)(217,271)
(218,273)(219,272)(220,274)(221,276)(222,275)(223,277)(224,279)(225,278)
(226,289)(227,291)(228,290)(229,292)(230,294)(231,293)(232,295)(233,297)
(234,296)(235,280)(236,282)(237,281)(238,283)(239,285)(240,284)(241,286)
(242,288)(243,287)(244,298)(245,300)(246,299)(247,301)(248,303)(249,302)
(250,304)(251,306)(252,305)(253,316)(254,318)(255,317)(256,319)(257,321)
(258,320)(259,322)(260,324)(261,323)(262,307)(263,309)(264,308)(265,310)
(266,312)(267,311)(268,313)(269,315)(270,314)(325,406)(326,408)(327,407)
(328,409)(329,411)(330,410)(331,412)(332,414)(333,413)(334,424)(335,426)
(336,425)(337,427)(338,429)(339,428)(340,430)(341,432)(342,431)(343,415)
(344,417)(345,416)(346,418)(347,420)(348,419)(349,421)(350,423)(351,422)
(352,379)(353,381)(354,380)(355,382)(356,384)(357,383)(358,385)(359,387)
(360,386)(361,397)(362,399)(363,398)(364,400)(365,402)(366,401)(367,403)
(368,405)(369,404)(370,388)(371,390)(372,389)(373,391)(374,393)(375,392)
(376,394)(377,396)(378,395);
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*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s1*s2*s0*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*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 >;
References : None.
to this polytope