Polytope of Type {12,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728f
if this polytope has a name.
Group : SmallGroup(1728,5273)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
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 : {12,12}*864c
3-fold quotients : {12,24}*576e
4-fold quotients : {6,12}*432b, {12,6}*432b
6-fold quotients : {12,12}*288a
8-fold quotients : {6,6}*216b
9-fold quotients : {4,24}*192b, {12,8}*192b
12-fold quotients : {6,12}*144a, {12,6}*144a
16-fold quotients : {6,6}*108
18-fold quotients : {4,12}*96a, {12,4}*96a
24-fold quotients : {6,6}*72a
27-fold quotients : {4,8}*64b
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
54-fold quotients : {4,4}*32
72-fold quotients : {2,6}*24, {6,2}*24
108-fold quotients : {2,4}*16, {4,2}*16
144-fold quotients : {2,3}*12, {3,2}*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)(109,361)(110,362)(111,363)(112,369)
(113,367)(114,368)(115,365)(116,366)(117,364)(118,352)(119,353)(120,354)
(121,360)(122,358)(123,359)(124,356)(125,357)(126,355)(127,370)(128,371)
(129,372)(130,378)(131,376)(132,377)(133,374)(134,375)(135,373)(136,334)
(137,335)(138,336)(139,342)(140,340)(141,341)(142,338)(143,339)(144,337)
(145,325)(146,326)(147,327)(148,333)(149,331)(150,332)(151,329)(152,330)
(153,328)(154,343)(155,344)(156,345)(157,351)(158,349)(159,350)(160,347)
(161,348)(162,346)(163,415)(164,416)(165,417)(166,423)(167,421)(168,422)
(169,419)(170,420)(171,418)(172,406)(173,407)(174,408)(175,414)(176,412)
(177,413)(178,410)(179,411)(180,409)(181,424)(182,425)(183,426)(184,432)
(185,430)(186,431)(187,428)(188,429)(189,427)(190,388)(191,389)(192,390)
(193,396)(194,394)(195,395)(196,392)(197,393)(198,391)(199,379)(200,380)
(201,381)(202,387)(203,385)(204,386)(205,383)(206,384)(207,382)(208,397)
(209,398)(210,399)(211,405)(212,403)(213,404)(214,401)(215,402)(216,400);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)
( 63, 89)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)( 70, 97)
( 71, 99)( 72, 98)( 73,103)( 74,105)( 75,104)( 76,100)( 77,102)( 78,101)
( 79,106)( 80,108)( 81,107)(109,139)(110,141)(111,140)(112,136)(113,138)
(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)(121,145)
(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)(129,158)
(130,154)(131,156)(132,155)(133,160)(134,162)(135,161)(163,166)(164,168)
(165,167)(170,171)(172,175)(173,177)(174,176)(179,180)(181,184)(182,186)
(183,185)(188,189)(190,193)(191,195)(192,194)(197,198)(199,202)(200,204)
(201,203)(206,207)(208,211)(209,213)(210,212)(215,216)(217,328)(218,330)
(219,329)(220,325)(221,327)(222,326)(223,331)(224,333)(225,332)(226,337)
(227,339)(228,338)(229,334)(230,336)(231,335)(232,340)(233,342)(234,341)
(235,346)(236,348)(237,347)(238,343)(239,345)(240,344)(241,349)(242,351)
(243,350)(244,355)(245,357)(246,356)(247,352)(248,354)(249,353)(250,358)
(251,360)(252,359)(253,364)(254,366)(255,365)(256,361)(257,363)(258,362)
(259,367)(260,369)(261,368)(262,373)(263,375)(264,374)(265,370)(266,372)
(267,371)(268,376)(269,378)(270,377)(271,409)(272,411)(273,410)(274,406)
(275,408)(276,407)(277,412)(278,414)(279,413)(280,418)(281,420)(282,419)
(283,415)(284,417)(285,416)(286,421)(287,423)(288,422)(289,427)(290,429)
(291,428)(292,424)(293,426)(294,425)(295,430)(296,432)(297,431)(298,382)
(299,384)(300,383)(301,379)(302,381)(303,380)(304,385)(305,387)(306,386)
(307,391)(308,393)(309,392)(310,388)(311,390)(312,389)(313,394)(314,396)
(315,395)(316,400)(317,402)(318,401)(319,397)(320,399)(321,398)(322,403)
(323,405)(324,404);;
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, s0*s1*s2*s1*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*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
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)(109,361)(110,362)(111,363)
(112,369)(113,367)(114,368)(115,365)(116,366)(117,364)(118,352)(119,353)
(120,354)(121,360)(122,358)(123,359)(124,356)(125,357)(126,355)(127,370)
(128,371)(129,372)(130,378)(131,376)(132,377)(133,374)(134,375)(135,373)
(136,334)(137,335)(138,336)(139,342)(140,340)(141,341)(142,338)(143,339)
(144,337)(145,325)(146,326)(147,327)(148,333)(149,331)(150,332)(151,329)
(152,330)(153,328)(154,343)(155,344)(156,345)(157,351)(158,349)(159,350)
(160,347)(161,348)(162,346)(163,415)(164,416)(165,417)(166,423)(167,421)
(168,422)(169,419)(170,420)(171,418)(172,406)(173,407)(174,408)(175,414)
(176,412)(177,413)(178,410)(179,411)(180,409)(181,424)(182,425)(183,426)
(184,432)(185,430)(186,431)(187,428)(188,429)(189,427)(190,388)(191,389)
(192,390)(193,396)(194,394)(195,395)(196,392)(197,393)(198,391)(199,379)
(200,380)(201,381)(202,387)(203,385)(204,386)(205,383)(206,384)(207,382)
(208,397)(209,398)(210,399)(211,405)(212,403)(213,404)(214,401)(215,402)
(216,400);
s2 := Sym(432)!( 1, 4)( 2, 6)( 3, 5)( 8, 9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)
( 62, 90)( 63, 89)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)
( 70, 97)( 71, 99)( 72, 98)( 73,103)( 74,105)( 75,104)( 76,100)( 77,102)
( 78,101)( 79,106)( 80,108)( 81,107)(109,139)(110,141)(111,140)(112,136)
(113,138)(114,137)(115,142)(116,144)(117,143)(118,148)(119,150)(120,149)
(121,145)(122,147)(123,146)(124,151)(125,153)(126,152)(127,157)(128,159)
(129,158)(130,154)(131,156)(132,155)(133,160)(134,162)(135,161)(163,166)
(164,168)(165,167)(170,171)(172,175)(173,177)(174,176)(179,180)(181,184)
(182,186)(183,185)(188,189)(190,193)(191,195)(192,194)(197,198)(199,202)
(200,204)(201,203)(206,207)(208,211)(209,213)(210,212)(215,216)(217,328)
(218,330)(219,329)(220,325)(221,327)(222,326)(223,331)(224,333)(225,332)
(226,337)(227,339)(228,338)(229,334)(230,336)(231,335)(232,340)(233,342)
(234,341)(235,346)(236,348)(237,347)(238,343)(239,345)(240,344)(241,349)
(242,351)(243,350)(244,355)(245,357)(246,356)(247,352)(248,354)(249,353)
(250,358)(251,360)(252,359)(253,364)(254,366)(255,365)(256,361)(257,363)
(258,362)(259,367)(260,369)(261,368)(262,373)(263,375)(264,374)(265,370)
(266,372)(267,371)(268,376)(269,378)(270,377)(271,409)(272,411)(273,410)
(274,406)(275,408)(276,407)(277,412)(278,414)(279,413)(280,418)(281,420)
(282,419)(283,415)(284,417)(285,416)(286,421)(287,423)(288,422)(289,427)
(290,429)(291,428)(292,424)(293,426)(294,425)(295,430)(296,432)(297,431)
(298,382)(299,384)(300,383)(301,379)(302,381)(303,380)(304,385)(305,387)
(306,386)(307,391)(308,393)(309,392)(310,388)(311,390)(312,389)(313,394)
(314,396)(315,395)(316,400)(317,402)(318,401)(319,397)(320,399)(321,398)
(322,403)(323,405)(324,404);
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, s0*s1*s2*s1*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*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;
References : None.
to this polytope