Polytope of Type {12,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,24}*1728b
if this polytope has a name.
Group : SmallGroup(1728,4110)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 36, 432, 72
Order of s₀s₁s₂ : 24
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 : {12,12}*864a, {6,24}*864c
   3-fold quotients : {12,24}*576b
   4-fold quotients : {12,6}*432a, {6,12}*432c
   6-fold quotients : {12,12}*288c, {6,24}*288c
   8-fold quotients : {6,6}*216c
   9-fold quotients : {12,8}*192a
   12-fold quotients : {12,6}*144b, {6,12}*144c
   16-fold quotients : {3,6}*108
   18-fold quotients : {12,4}*96a, {6,8}*96
   24-fold quotients : {6,6}*72c
   27-fold quotients : {4,8}*64a
   36-fold quotients : {12,2}*48, {6,4}*48a
   48-fold quotients : {3,6}*36
   54-fold quotients : {4,4}*32, {2,8}*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)
( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)( 68, 80)
( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)( 91,100)
( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)( 99,105)
(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)(122,134)
(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)(145,154)
(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)(153,159)
(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)( 55,320)( 56,321)
( 57,319)( 58,318)( 59,316)( 60,317)( 61,322)( 62,323)( 63,324)( 64,311)
( 65,312)( 66,310)( 67,309)( 68,307)( 69,308)( 70,313)( 71,314)( 72,315)
( 73,302)( 74,303)( 75,301)( 76,300)( 77,298)( 78,299)( 79,304)( 80,305)
( 81,306)( 82,293)( 83,294)( 84,292)( 85,291)( 86,289)( 87,290)( 88,295)
( 89,296)( 90,297)( 91,284)( 92,285)( 93,283)( 94,282)( 95,280)( 96,281)
( 97,286)( 98,287)( 99,288)(100,275)(101,276)(102,274)(103,273)(104,271)
(105,272)(106,277)(107,278)(108,279)(109,374)(110,375)(111,373)(112,372)
(113,370)(114,371)(115,376)(116,377)(117,378)(118,365)(119,366)(120,364)
(121,363)(122,361)(123,362)(124,367)(125,368)(126,369)(127,356)(128,357)
(129,355)(130,354)(131,352)(132,353)(133,358)(134,359)(135,360)(136,347)
(137,348)(138,346)(139,345)(140,343)(141,344)(142,349)(143,350)(144,351)
(145,338)(146,339)(147,337)(148,336)(149,334)(150,335)(151,340)(152,341)
(153,342)(154,329)(155,330)(156,328)(157,327)(158,325)(159,326)(160,331)
(161,332)(162,333)(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)
( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)
( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)( 70,106)
( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)
( 79, 97)( 80, 99)( 81, 98)(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)(164,165)(167,168)
(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)(177,185)(178,187)
(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)(200,210)(201,209)
(202,211)(203,213)(204,212)(205,214)(206,216)(207,215)(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*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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)( 58, 61)( 59, 62)( 60, 63)( 64, 73)( 65, 74)( 66, 75)( 67, 79)
( 68, 80)( 69, 81)( 70, 76)( 71, 77)( 72, 78)( 85, 88)( 86, 89)( 87, 90)
( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)( 97,103)( 98,104)
( 99,105)(112,115)(113,116)(114,117)(118,127)(119,128)(120,129)(121,133)
(122,134)(123,135)(124,130)(125,131)(126,132)(139,142)(140,143)(141,144)
(145,154)(146,155)(147,156)(148,160)(149,161)(150,162)(151,157)(152,158)
(153,159)(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)( 55,320)
( 56,321)( 57,319)( 58,318)( 59,316)( 60,317)( 61,322)( 62,323)( 63,324)
( 64,311)( 65,312)( 66,310)( 67,309)( 68,307)( 69,308)( 70,313)( 71,314)
( 72,315)( 73,302)( 74,303)( 75,301)( 76,300)( 77,298)( 78,299)( 79,304)
( 80,305)( 81,306)( 82,293)( 83,294)( 84,292)( 85,291)( 86,289)( 87,290)
( 88,295)( 89,296)( 90,297)( 91,284)( 92,285)( 93,283)( 94,282)( 95,280)
( 96,281)( 97,286)( 98,287)( 99,288)(100,275)(101,276)(102,274)(103,273)
(104,271)(105,272)(106,277)(107,278)(108,279)(109,374)(110,375)(111,373)
(112,372)(113,370)(114,371)(115,376)(116,377)(117,378)(118,365)(119,366)
(120,364)(121,363)(122,361)(123,362)(124,367)(125,368)(126,369)(127,356)
(128,357)(129,355)(130,354)(131,352)(132,353)(133,358)(134,359)(135,360)
(136,347)(137,348)(138,346)(139,345)(140,343)(141,344)(142,349)(143,350)
(144,351)(145,338)(146,339)(147,337)(148,336)(149,334)(150,335)(151,340)
(152,341)(153,342)(154,329)(155,330)(156,328)(157,327)(158,325)(159,326)
(160,331)(161,332)(162,333)(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)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)
( 62, 90)( 63, 89)( 64,100)( 65,102)( 66,101)( 67,103)( 68,105)( 69,104)
( 70,106)( 71,108)( 72,107)( 73, 91)( 74, 93)( 75, 92)( 76, 94)( 77, 96)
( 78, 95)( 79, 97)( 80, 99)( 81, 98)(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)(164,165)
(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)(177,185)
(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)(200,210)
(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215)(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*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*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope