Polytope of Type {12,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728b
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
Self-Petrie
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}*864b
3-fold quotients : {12,12}*576b
4-fold quotients : {6,12}*432a, {12,6}*432c
6-fold quotients : {12,12}*288b
8-fold quotients : {6,6}*216a
9-fold quotients : {4,12}*192a
12-fold quotients : {6,12}*144b, {12,6}*144c
16-fold quotients : {6,3}*108
18-fold quotients : {4,12}*96a
24-fold quotients : {6,6}*72b
27-fold quotients : {4,4}*64
36-fold quotients : {2,12}*48, {4,6}*48a
48-fold quotients : {6,3}*36
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24
108-fold quotients : {2,4}*16, {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 := ( 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);;
s1 := ( 1,226)( 2,227)( 3,228)( 4,234)( 5,232)( 6,233)( 7,230)( 8,231)
( 9,229)( 10,217)( 11,218)( 12,219)( 13,225)( 14,223)( 15,224)( 16,221)
( 17,222)( 18,220)( 19,235)( 20,236)( 21,237)( 22,243)( 23,241)( 24,242)
( 25,239)( 26,240)( 27,238)( 28,253)( 29,254)( 30,255)( 31,261)( 32,259)
( 33,260)( 34,257)( 35,258)( 36,256)( 37,244)( 38,245)( 39,246)( 40,252)
( 41,250)( 42,251)( 43,248)( 44,249)( 45,247)( 46,262)( 47,263)( 48,264)
( 49,270)( 50,268)( 51,269)( 52,266)( 53,267)( 54,265)( 55,280)( 56,281)
( 57,282)( 58,288)( 59,286)( 60,287)( 61,284)( 62,285)( 63,283)( 64,271)
( 65,272)( 66,273)( 67,279)( 68,277)( 69,278)( 70,275)( 71,276)( 72,274)
( 73,289)( 74,290)( 75,291)( 76,297)( 77,295)( 78,296)( 79,293)( 80,294)
( 81,292)( 82,307)( 83,308)( 84,309)( 85,315)( 86,313)( 87,314)( 88,311)
( 89,312)( 90,310)( 91,298)( 92,299)( 93,300)( 94,306)( 95,304)( 96,305)
( 97,302)( 98,303)( 99,301)(100,316)(101,317)(102,318)(103,324)(104,322)
(105,323)(106,320)(107,321)(108,319)(109,334)(110,335)(111,336)(112,342)
(113,340)(114,341)(115,338)(116,339)(117,337)(118,325)(119,326)(120,327)
(121,333)(122,331)(123,332)(124,329)(125,330)(126,328)(127,343)(128,344)
(129,345)(130,351)(131,349)(132,350)(133,347)(134,348)(135,346)(136,361)
(137,362)(138,363)(139,369)(140,367)(141,368)(142,365)(143,366)(144,364)
(145,352)(146,353)(147,354)(148,360)(149,358)(150,359)(151,356)(152,357)
(153,355)(154,370)(155,371)(156,372)(157,378)(158,376)(159,377)(160,374)
(161,375)(162,373)(163,388)(164,389)(165,390)(166,396)(167,394)(168,395)
(169,392)(170,393)(171,391)(172,379)(173,380)(174,381)(175,387)(176,385)
(177,386)(178,383)(179,384)(180,382)(181,397)(182,398)(183,399)(184,405)
(185,403)(186,404)(187,401)(188,402)(189,400)(190,415)(191,416)(192,417)
(193,423)(194,421)(195,422)(196,419)(197,420)(198,418)(199,406)(200,407)
(201,408)(202,414)(203,412)(204,413)(205,410)(206,411)(207,409)(208,424)
(209,425)(210,426)(211,432)(212,430)(213,431)(214,428)(215,429)(216,427);;
s2 := ( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)( 61, 88)( 62, 89)
( 63, 90)( 64,103)( 65,104)( 66,105)( 67,100)( 68,101)( 69,102)( 70,106)
( 71,107)( 72,108)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)( 78, 93)
( 79, 97)( 80, 98)( 81, 99)(109,139)(110,140)(111,141)(112,136)(113,137)
(114,138)(115,142)(116,143)(117,144)(118,157)(119,158)(120,159)(121,154)
(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)(129,150)
(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(163,166)(164,167)
(165,168)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)(178,187)
(179,188)(180,189)(190,193)(191,194)(192,195)(199,211)(200,212)(201,213)
(202,208)(203,209)(204,210)(205,214)(206,215)(207,216)(217,409)(218,410)
(219,411)(220,406)(221,407)(222,408)(223,412)(224,413)(225,414)(226,427)
(227,428)(228,429)(229,424)(230,425)(231,426)(232,430)(233,431)(234,432)
(235,418)(236,419)(237,420)(238,415)(239,416)(240,417)(241,421)(242,422)
(243,423)(244,382)(245,383)(246,384)(247,379)(248,380)(249,381)(250,385)
(251,386)(252,387)(253,400)(254,401)(255,402)(256,397)(257,398)(258,399)
(259,403)(260,404)(261,405)(262,391)(263,392)(264,393)(265,388)(266,389)
(267,390)(268,394)(269,395)(270,396)(271,328)(272,329)(273,330)(274,325)
(275,326)(276,327)(277,331)(278,332)(279,333)(280,346)(281,347)(282,348)
(283,343)(284,344)(285,345)(286,349)(287,350)(288,351)(289,337)(290,338)
(291,339)(292,334)(293,335)(294,336)(295,340)(296,341)(297,342)(298,355)
(299,356)(300,357)(301,352)(302,353)(303,354)(304,358)(305,359)(306,360)
(307,373)(308,374)(309,375)(310,370)(311,371)(312,372)(313,376)(314,377)
(315,378)(316,364)(317,365)(318,366)(319,361)(320,362)(321,363)(322,367)
(323,368)(324,369);;
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*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,
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)!( 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);
s1 := Sym(432)!( 1,226)( 2,227)( 3,228)( 4,234)( 5,232)( 6,233)( 7,230)
( 8,231)( 9,229)( 10,217)( 11,218)( 12,219)( 13,225)( 14,223)( 15,224)
( 16,221)( 17,222)( 18,220)( 19,235)( 20,236)( 21,237)( 22,243)( 23,241)
( 24,242)( 25,239)( 26,240)( 27,238)( 28,253)( 29,254)( 30,255)( 31,261)
( 32,259)( 33,260)( 34,257)( 35,258)( 36,256)( 37,244)( 38,245)( 39,246)
( 40,252)( 41,250)( 42,251)( 43,248)( 44,249)( 45,247)( 46,262)( 47,263)
( 48,264)( 49,270)( 50,268)( 51,269)( 52,266)( 53,267)( 54,265)( 55,280)
( 56,281)( 57,282)( 58,288)( 59,286)( 60,287)( 61,284)( 62,285)( 63,283)
( 64,271)( 65,272)( 66,273)( 67,279)( 68,277)( 69,278)( 70,275)( 71,276)
( 72,274)( 73,289)( 74,290)( 75,291)( 76,297)( 77,295)( 78,296)( 79,293)
( 80,294)( 81,292)( 82,307)( 83,308)( 84,309)( 85,315)( 86,313)( 87,314)
( 88,311)( 89,312)( 90,310)( 91,298)( 92,299)( 93,300)( 94,306)( 95,304)
( 96,305)( 97,302)( 98,303)( 99,301)(100,316)(101,317)(102,318)(103,324)
(104,322)(105,323)(106,320)(107,321)(108,319)(109,334)(110,335)(111,336)
(112,342)(113,340)(114,341)(115,338)(116,339)(117,337)(118,325)(119,326)
(120,327)(121,333)(122,331)(123,332)(124,329)(125,330)(126,328)(127,343)
(128,344)(129,345)(130,351)(131,349)(132,350)(133,347)(134,348)(135,346)
(136,361)(137,362)(138,363)(139,369)(140,367)(141,368)(142,365)(143,366)
(144,364)(145,352)(146,353)(147,354)(148,360)(149,358)(150,359)(151,356)
(152,357)(153,355)(154,370)(155,371)(156,372)(157,378)(158,376)(159,377)
(160,374)(161,375)(162,373)(163,388)(164,389)(165,390)(166,396)(167,394)
(168,395)(169,392)(170,393)(171,391)(172,379)(173,380)(174,381)(175,387)
(176,385)(177,386)(178,383)(179,384)(180,382)(181,397)(182,398)(183,399)
(184,405)(185,403)(186,404)(187,401)(188,402)(189,400)(190,415)(191,416)
(192,417)(193,423)(194,421)(195,422)(196,419)(197,420)(198,418)(199,406)
(200,407)(201,408)(202,414)(203,412)(204,413)(205,410)(206,411)(207,409)
(208,424)(209,425)(210,426)(211,432)(212,430)(213,431)(214,428)(215,429)
(216,427);
s2 := Sym(432)!( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 85)( 56, 86)( 57, 87)( 58, 82)( 59, 83)( 60, 84)( 61, 88)
( 62, 89)( 63, 90)( 64,103)( 65,104)( 66,105)( 67,100)( 68,101)( 69,102)
( 70,106)( 71,107)( 72,108)( 73, 94)( 74, 95)( 75, 96)( 76, 91)( 77, 92)
( 78, 93)( 79, 97)( 80, 98)( 81, 99)(109,139)(110,140)(111,141)(112,136)
(113,137)(114,138)(115,142)(116,143)(117,144)(118,157)(119,158)(120,159)
(121,154)(122,155)(123,156)(124,160)(125,161)(126,162)(127,148)(128,149)
(129,150)(130,145)(131,146)(132,147)(133,151)(134,152)(135,153)(163,166)
(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)(176,182)(177,183)
(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)(199,211)(200,212)
(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)(207,216)(217,409)
(218,410)(219,411)(220,406)(221,407)(222,408)(223,412)(224,413)(225,414)
(226,427)(227,428)(228,429)(229,424)(230,425)(231,426)(232,430)(233,431)
(234,432)(235,418)(236,419)(237,420)(238,415)(239,416)(240,417)(241,421)
(242,422)(243,423)(244,382)(245,383)(246,384)(247,379)(248,380)(249,381)
(250,385)(251,386)(252,387)(253,400)(254,401)(255,402)(256,397)(257,398)
(258,399)(259,403)(260,404)(261,405)(262,391)(263,392)(264,393)(265,388)
(266,389)(267,390)(268,394)(269,395)(270,396)(271,328)(272,329)(273,330)
(274,325)(275,326)(276,327)(277,331)(278,332)(279,333)(280,346)(281,347)
(282,348)(283,343)(284,344)(285,345)(286,349)(287,350)(288,351)(289,337)
(290,338)(291,339)(292,334)(293,335)(294,336)(295,340)(296,341)(297,342)
(298,355)(299,356)(300,357)(301,352)(302,353)(303,354)(304,358)(305,359)
(306,360)(307,373)(308,374)(309,375)(310,370)(311,371)(312,372)(313,376)
(314,377)(315,378)(316,364)(317,365)(318,366)(319,361)(320,362)(321,363)
(322,367)(323,368)(324,369);
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*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s2*s0*s1*s0,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*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,
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