Polytope of Type {24,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,3511)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 72, 432, 36
Order of s0s1s2 : 24
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 : {24,12}*576a
4-fold quotients : {6,12}*432a, {12,6}*432c
6-fold quotients : {12,12}*288b
8-fold quotients : {6,6}*216a
9-fold quotients : {8,12}*192b
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 : {8,4}*64b
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)
( 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)(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)(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)(325,379)(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);;
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, 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)(217,382)(218,383)
(219,384)(220,379)(221,380)(222,381)(223,385)(224,386)(225,387)(226,400)
(227,401)(228,402)(229,397)(230,398)(231,399)(232,403)(233,404)(234,405)
(235,391)(236,392)(237,393)(238,388)(239,389)(240,390)(241,394)(242,395)
(243,396)(244,409)(245,410)(246,411)(247,406)(248,407)(249,408)(250,412)
(251,413)(252,414)(253,427)(254,428)(255,429)(256,424)(257,425)(258,426)
(259,430)(260,431)(261,432)(262,418)(263,419)(264,420)(265,415)(266,416)
(267,417)(268,421)(269,422)(270,423)(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*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*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)(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)(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)(325,379)(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);
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, 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)(217,382)
(218,383)(219,384)(220,379)(221,380)(222,381)(223,385)(224,386)(225,387)
(226,400)(227,401)(228,402)(229,397)(230,398)(231,399)(232,403)(233,404)
(234,405)(235,391)(236,392)(237,393)(238,388)(239,389)(240,390)(241,394)
(242,395)(243,396)(244,409)(245,410)(246,411)(247,406)(248,407)(249,408)
(250,412)(251,413)(252,414)(253,427)(254,428)(255,429)(256,424)(257,425)
(258,426)(259,430)(260,431)(261,432)(262,418)(263,419)(264,420)(265,415)
(266,416)(267,417)(268,421)(269,422)(270,423)(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*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*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