Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,5113)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s₀s₁s₂ : 24
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 : {24,6}*864a, {12,12}*864a
   3-fold quotients : {24,12}*576d
   4-fold quotients : {12,6}*432a, {6,12}*432c
   6-fold quotients : {24,6}*288b, {12,12}*288c
   8-fold quotients : {6,6}*216c
   9-fold quotients : {24,4}*192a
   12-fold quotients : {12,6}*144b, {6,12}*144c
   16-fold quotients : {3,6}*108
   18-fold quotients : {12,4}*96a, {24,2}*96
   24-fold quotients : {6,6}*72c
   27-fold quotients : {8,4}*64a
   36-fold quotients : {12,2}*48, {6,4}*48a
   48-fold quotients : {3,6}*36
   54-fold quotients : {4,4}*32, {8,2}*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)
(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)(163,190)(164,191)(165,192)(166,196)(167,197)
(168,198)(169,193)(170,194)(171,195)(172,208)(173,209)(174,210)(175,214)
(176,215)(177,216)(178,211)(179,212)(180,213)(181,199)(182,200)(183,201)
(184,205)(185,206)(186,207)(187,202)(188,203)(189,204)(217,379)(218,380)
(219,381)(220,385)(221,386)(222,387)(223,382)(224,383)(225,384)(226,397)
(227,398)(228,399)(229,403)(230,404)(231,405)(232,400)(233,401)(234,402)
(235,388)(236,389)(237,390)(238,394)(239,395)(240,396)(241,391)(242,392)
(243,393)(244,406)(245,407)(246,408)(247,412)(248,413)(249,414)(250,409)
(251,410)(252,411)(253,424)(254,425)(255,426)(256,430)(257,431)(258,432)
(259,427)(260,428)(261,429)(262,415)(263,416)(264,417)(265,421)(266,422)
(267,423)(268,418)(269,419)(270,420)(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)(109,374)(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)(163,428)(164,429)(165,427)(166,426)(167,424)(168,425)
(169,430)(170,431)(171,432)(172,419)(173,420)(174,418)(175,417)(176,415)
(177,416)(178,421)(179,422)(180,423)(181,410)(182,411)(183,409)(184,408)
(185,406)(186,407)(187,412)(188,413)(189,414)(190,401)(191,402)(192,400)
(193,399)(194,397)(195,398)(196,403)(197,404)(198,405)(199,392)(200,393)
(201,391)(202,390)(203,388)(204,389)(205,394)(206,395)(207,396)(208,383)
(209,384)(210,382)(211,381)(212,379)(213,380)(214,385)(215,386)(216,387);;
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)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(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)(325,379)(326,381)
(327,380)(328,382)(329,384)(330,383)(331,385)(332,387)(333,386)(334,397)
(335,399)(336,398)(337,400)(338,402)(339,401)(340,403)(341,405)(342,404)
(343,388)(344,390)(345,389)(346,391)(347,393)(348,392)(349,394)(350,396)
(351,395)(352,406)(353,408)(354,407)(355,409)(356,411)(357,410)(358,412)
(359,414)(360,413)(361,424)(362,426)(363,425)(364,427)(365,429)(366,428)
(367,430)(368,432)(369,431)(370,415)(371,417)(372,416)(373,418)(374,420)
(375,419)(376,421)(377,423)(378,422);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*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, 
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*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)( 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)(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)(163,190)(164,191)(165,192)(166,196)
(167,197)(168,198)(169,193)(170,194)(171,195)(172,208)(173,209)(174,210)
(175,214)(176,215)(177,216)(178,211)(179,212)(180,213)(181,199)(182,200)
(183,201)(184,205)(185,206)(186,207)(187,202)(188,203)(189,204)(217,379)
(218,380)(219,381)(220,385)(221,386)(222,387)(223,382)(224,383)(225,384)
(226,397)(227,398)(228,399)(229,403)(230,404)(231,405)(232,400)(233,401)
(234,402)(235,388)(236,389)(237,390)(238,394)(239,395)(240,396)(241,391)
(242,392)(243,393)(244,406)(245,407)(246,408)(247,412)(248,413)(249,414)
(250,409)(251,410)(252,411)(253,424)(254,425)(255,426)(256,430)(257,431)
(258,432)(259,427)(260,428)(261,429)(262,415)(263,416)(264,417)(265,421)
(266,422)(267,423)(268,418)(269,419)(270,420)(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)(109,374)(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)(163,428)(164,429)(165,427)(166,426)(167,424)
(168,425)(169,430)(170,431)(171,432)(172,419)(173,420)(174,418)(175,417)
(176,415)(177,416)(178,421)(179,422)(180,423)(181,410)(182,411)(183,409)
(184,408)(185,406)(186,407)(187,412)(188,413)(189,414)(190,401)(191,402)
(192,400)(193,399)(194,397)(195,398)(196,403)(197,404)(198,405)(199,392)
(200,393)(201,391)(202,390)(203,388)(204,389)(205,394)(206,395)(207,396)
(208,383)(209,384)(210,382)(211,381)(212,379)(213,380)(214,385)(215,386)
(216,387);
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)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(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)(325,379)
(326,381)(327,380)(328,382)(329,384)(330,383)(331,385)(332,387)(333,386)
(334,397)(335,399)(336,398)(337,400)(338,402)(339,401)(340,403)(341,405)
(342,404)(343,388)(344,390)(345,389)(346,391)(347,393)(348,392)(349,394)
(350,396)(351,395)(352,406)(353,408)(354,407)(355,409)(356,411)(357,410)
(358,412)(359,414)(360,413)(361,424)(362,426)(363,425)(364,427)(365,429)
(366,428)(367,430)(368,432)(369,431)(370,415)(371,417)(372,416)(373,418)
(374,420)(375,419)(376,421)(377,423)(378,422);
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*s0*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*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, 
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope