Polytope of Type {24,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,4110)
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
   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, {24,6}*864c
   3-fold quotients : {24,12}*576b
   4-fold quotients : {6,12}*432a, {12,6}*432c
   6-fold quotients : {12,12}*288b, {24,6}*288c
   8-fold quotients : {6,6}*216a
   9-fold quotients : {8,12}*192a
   12-fold quotients : {6,12}*144b, {12,6}*144c
   16-fold quotients : {6,3}*108
   18-fold quotients : {4,12}*96a, {8,6}*96
   24-fold quotients : {6,6}*72b
   27-fold quotients : {8,4}*64a
   36-fold quotients : {2,12}*48, {4,6}*48a
   48-fold quotients : {6,3}*36
   54-fold quotients : {4,4}*32, {8,2}*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)
( 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);;
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)( 55,307)( 56,308)
( 57,309)( 58,315)( 59,313)( 60,314)( 61,311)( 62,312)( 63,310)( 64,298)
( 65,299)( 66,300)( 67,306)( 68,304)( 69,305)( 70,302)( 71,303)( 72,301)
( 73,316)( 74,317)( 75,318)( 76,324)( 77,322)( 78,323)( 79,320)( 80,321)
( 81,319)( 82,280)( 83,281)( 84,282)( 85,288)( 86,286)( 87,287)( 88,284)
( 89,285)( 90,283)( 91,271)( 92,272)( 93,273)( 94,279)( 95,277)( 96,278)
( 97,275)( 98,276)( 99,274)(100,289)(101,290)(102,291)(103,297)(104,295)
(105,296)(106,293)(107,294)(108,292)(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)(163,388)(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, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)( 68, 74)
( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)( 91,103)
( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)( 99,108)
(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)(122,128)
(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)(145,157)
(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)
(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, 
s2*s1*s2*s1*s0*s1*s2*s1*s2*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)( 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);
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)( 55,307)
( 56,308)( 57,309)( 58,315)( 59,313)( 60,314)( 61,311)( 62,312)( 63,310)
( 64,298)( 65,299)( 66,300)( 67,306)( 68,304)( 69,305)( 70,302)( 71,303)
( 72,301)( 73,316)( 74,317)( 75,318)( 76,324)( 77,322)( 78,323)( 79,320)
( 80,321)( 81,319)( 82,280)( 83,281)( 84,282)( 85,288)( 86,286)( 87,287)
( 88,284)( 89,285)( 90,283)( 91,271)( 92,272)( 93,273)( 94,279)( 95,277)
( 96,278)( 97,275)( 98,276)( 99,274)(100,289)(101,290)(102,291)(103,297)
(104,295)(105,296)(106,293)(107,294)(108,292)(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)(163,388)(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, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)
( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)
( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)
( 99,108)(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)
(122,128)(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)
(145,157)(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)
(153,162)(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, 
s2*s1*s2*s1*s0*s1*s2*s1*s2*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