Polytope of Type {24,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,12}*1728f
if this polytope has a name.
Group : SmallGroup(1728,5273)
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
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}*864c
3-fold quotients : {24,12}*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 : {24,4}*192b, {8,12}*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 : {8,4}*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)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)
( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)
( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)(166,169)
(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)(184,187)
(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)(202,205)
(203,207)(204,206)(209,210)(211,214)(212,216)(213,215)(217,325)(218,327)
(219,326)(220,331)(221,333)(222,332)(223,328)(224,330)(225,329)(226,334)
(227,336)(228,335)(229,340)(230,342)(231,341)(232,337)(233,339)(234,338)
(235,343)(236,345)(237,344)(238,349)(239,351)(240,350)(241,346)(242,348)
(243,347)(244,352)(245,354)(246,353)(247,358)(248,360)(249,359)(250,355)
(251,357)(252,356)(253,361)(254,363)(255,362)(256,367)(257,369)(258,368)
(259,364)(260,366)(261,365)(262,370)(263,372)(264,371)(265,376)(266,378)
(267,377)(268,373)(269,375)(270,374)(271,406)(272,408)(273,407)(274,412)
(275,414)(276,413)(277,409)(278,411)(279,410)(280,415)(281,417)(282,416)
(283,421)(284,423)(285,422)(286,418)(287,420)(288,419)(289,424)(290,426)
(291,425)(292,430)(293,432)(294,431)(295,427)(296,429)(297,428)(298,379)
(299,381)(300,380)(301,385)(302,387)(303,386)(304,382)(305,384)(306,383)
(307,388)(308,390)(309,389)(310,394)(311,396)(312,395)(313,391)(314,393)
(315,392)(316,397)(317,399)(318,398)(319,403)(320,405)(321,404)(322,400)
(323,402)(324,401);;
s1 := ( 1,220)( 2,221)( 3,222)( 4,217)( 5,218)( 6,219)( 7,223)( 8,224)
( 9,225)( 10,238)( 11,239)( 12,240)( 13,235)( 14,236)( 15,237)( 16,241)
( 17,242)( 18,243)( 19,229)( 20,230)( 21,231)( 22,226)( 23,227)( 24,228)
( 25,232)( 26,233)( 27,234)( 28,247)( 29,248)( 30,249)( 31,244)( 32,245)
( 33,246)( 34,250)( 35,251)( 36,252)( 37,265)( 38,266)( 39,267)( 40,262)
( 41,263)( 42,264)( 43,268)( 44,269)( 45,270)( 46,256)( 47,257)( 48,258)
( 49,253)( 50,254)( 51,255)( 52,259)( 53,260)( 54,261)( 55,274)( 56,275)
( 57,276)( 58,271)( 59,272)( 60,273)( 61,277)( 62,278)( 63,279)( 64,292)
( 65,293)( 66,294)( 67,289)( 68,290)( 69,291)( 70,295)( 71,296)( 72,297)
( 73,283)( 74,284)( 75,285)( 76,280)( 77,281)( 78,282)( 79,286)( 80,287)
( 81,288)( 82,301)( 83,302)( 84,303)( 85,298)( 86,299)( 87,300)( 88,304)
( 89,305)( 90,306)( 91,319)( 92,320)( 93,321)( 94,316)( 95,317)( 96,318)
( 97,322)( 98,323)( 99,324)(100,310)(101,311)(102,312)(103,307)(104,308)
(105,309)(106,313)(107,314)(108,315)(109,355)(110,356)(111,357)(112,352)
(113,353)(114,354)(115,358)(116,359)(117,360)(118,373)(119,374)(120,375)
(121,370)(122,371)(123,372)(124,376)(125,377)(126,378)(127,364)(128,365)
(129,366)(130,361)(131,362)(132,363)(133,367)(134,368)(135,369)(136,328)
(137,329)(138,330)(139,325)(140,326)(141,327)(142,331)(143,332)(144,333)
(145,346)(146,347)(147,348)(148,343)(149,344)(150,345)(151,349)(152,350)
(153,351)(154,337)(155,338)(156,339)(157,334)(158,335)(159,336)(160,340)
(161,341)(162,342)(163,409)(164,410)(165,411)(166,406)(167,407)(168,408)
(169,412)(170,413)(171,414)(172,427)(173,428)(174,429)(175,424)(176,425)
(177,426)(178,430)(179,431)(180,432)(181,418)(182,419)(183,420)(184,415)
(185,416)(186,417)(187,421)(188,422)(189,423)(190,382)(191,383)(192,384)
(193,379)(194,380)(195,381)(196,385)(197,386)(198,387)(199,400)(200,401)
(201,402)(202,397)(203,398)(204,399)(205,403)(206,404)(207,405)(208,391)
(209,392)(210,393)(211,388)(212,389)(213,390)(214,394)(215,395)(216,396);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)( 8, 17)
( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)( 62, 71)
( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)( 85, 95)
( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)(106,108)
(109,145)(110,147)(111,146)(112,149)(113,148)(114,150)(115,153)(116,152)
(117,151)(118,136)(119,138)(120,137)(121,140)(122,139)(123,141)(124,144)
(125,143)(126,142)(127,154)(128,156)(129,155)(130,158)(131,157)(132,159)
(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,203)(167,202)
(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)(175,194)
(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)(183,209)
(184,212)(185,211)(186,213)(187,216)(188,215)(189,214)(217,280)(218,282)
(219,281)(220,284)(221,283)(222,285)(223,288)(224,287)(225,286)(226,271)
(227,273)(228,272)(229,275)(230,274)(231,276)(232,279)(233,278)(234,277)
(235,289)(236,291)(237,290)(238,293)(239,292)(240,294)(241,297)(242,296)
(243,295)(244,307)(245,309)(246,308)(247,311)(248,310)(249,312)(250,315)
(251,314)(252,313)(253,298)(254,300)(255,299)(256,302)(257,301)(258,303)
(259,306)(260,305)(261,304)(262,316)(263,318)(264,317)(265,320)(266,319)
(267,321)(268,324)(269,323)(270,322)(325,415)(326,417)(327,416)(328,419)
(329,418)(330,420)(331,423)(332,422)(333,421)(334,406)(335,408)(336,407)
(337,410)(338,409)(339,411)(340,414)(341,413)(342,412)(343,424)(344,426)
(345,425)(346,428)(347,427)(348,429)(349,432)(350,431)(351,430)(352,388)
(353,390)(354,389)(355,392)(356,391)(357,393)(358,396)(359,395)(360,394)
(361,379)(362,381)(363,380)(364,383)(365,382)(366,384)(367,387)(368,386)
(369,385)(370,397)(371,399)(372,398)(373,401)(374,400)(375,402)(376,405)
(377,404)(378,403);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)
( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)
( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)
( 78,107)( 79,103)( 80,105)( 81,104)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158)(164,165)
(166,169)(167,171)(168,170)(173,174)(175,178)(176,180)(177,179)(182,183)
(184,187)(185,189)(186,188)(191,192)(193,196)(194,198)(195,197)(200,201)
(202,205)(203,207)(204,206)(209,210)(211,214)(212,216)(213,215)(217,325)
(218,327)(219,326)(220,331)(221,333)(222,332)(223,328)(224,330)(225,329)
(226,334)(227,336)(228,335)(229,340)(230,342)(231,341)(232,337)(233,339)
(234,338)(235,343)(236,345)(237,344)(238,349)(239,351)(240,350)(241,346)
(242,348)(243,347)(244,352)(245,354)(246,353)(247,358)(248,360)(249,359)
(250,355)(251,357)(252,356)(253,361)(254,363)(255,362)(256,367)(257,369)
(258,368)(259,364)(260,366)(261,365)(262,370)(263,372)(264,371)(265,376)
(266,378)(267,377)(268,373)(269,375)(270,374)(271,406)(272,408)(273,407)
(274,412)(275,414)(276,413)(277,409)(278,411)(279,410)(280,415)(281,417)
(282,416)(283,421)(284,423)(285,422)(286,418)(287,420)(288,419)(289,424)
(290,426)(291,425)(292,430)(293,432)(294,431)(295,427)(296,429)(297,428)
(298,379)(299,381)(300,380)(301,385)(302,387)(303,386)(304,382)(305,384)
(306,383)(307,388)(308,390)(309,389)(310,394)(311,396)(312,395)(313,391)
(314,393)(315,392)(316,397)(317,399)(318,398)(319,403)(320,405)(321,404)
(322,400)(323,402)(324,401);
s1 := Sym(432)!( 1,220)( 2,221)( 3,222)( 4,217)( 5,218)( 6,219)( 7,223)
( 8,224)( 9,225)( 10,238)( 11,239)( 12,240)( 13,235)( 14,236)( 15,237)
( 16,241)( 17,242)( 18,243)( 19,229)( 20,230)( 21,231)( 22,226)( 23,227)
( 24,228)( 25,232)( 26,233)( 27,234)( 28,247)( 29,248)( 30,249)( 31,244)
( 32,245)( 33,246)( 34,250)( 35,251)( 36,252)( 37,265)( 38,266)( 39,267)
( 40,262)( 41,263)( 42,264)( 43,268)( 44,269)( 45,270)( 46,256)( 47,257)
( 48,258)( 49,253)( 50,254)( 51,255)( 52,259)( 53,260)( 54,261)( 55,274)
( 56,275)( 57,276)( 58,271)( 59,272)( 60,273)( 61,277)( 62,278)( 63,279)
( 64,292)( 65,293)( 66,294)( 67,289)( 68,290)( 69,291)( 70,295)( 71,296)
( 72,297)( 73,283)( 74,284)( 75,285)( 76,280)( 77,281)( 78,282)( 79,286)
( 80,287)( 81,288)( 82,301)( 83,302)( 84,303)( 85,298)( 86,299)( 87,300)
( 88,304)( 89,305)( 90,306)( 91,319)( 92,320)( 93,321)( 94,316)( 95,317)
( 96,318)( 97,322)( 98,323)( 99,324)(100,310)(101,311)(102,312)(103,307)
(104,308)(105,309)(106,313)(107,314)(108,315)(109,355)(110,356)(111,357)
(112,352)(113,353)(114,354)(115,358)(116,359)(117,360)(118,373)(119,374)
(120,375)(121,370)(122,371)(123,372)(124,376)(125,377)(126,378)(127,364)
(128,365)(129,366)(130,361)(131,362)(132,363)(133,367)(134,368)(135,369)
(136,328)(137,329)(138,330)(139,325)(140,326)(141,327)(142,331)(143,332)
(144,333)(145,346)(146,347)(147,348)(148,343)(149,344)(150,345)(151,349)
(152,350)(153,351)(154,337)(155,338)(156,339)(157,334)(158,335)(159,336)
(160,340)(161,341)(162,342)(163,409)(164,410)(165,411)(166,406)(167,407)
(168,408)(169,412)(170,413)(171,414)(172,427)(173,428)(174,429)(175,424)
(176,425)(177,426)(178,430)(179,431)(180,432)(181,418)(182,419)(183,420)
(184,415)(185,416)(186,417)(187,421)(188,422)(189,423)(190,382)(191,383)
(192,384)(193,379)(194,380)(195,381)(196,385)(197,386)(198,387)(199,400)
(200,401)(201,402)(202,397)(203,398)(204,399)(205,403)(206,404)(207,405)
(208,391)(209,392)(210,393)(211,388)(212,389)(213,390)(214,394)(215,395)
(216,396);
s2 := Sym(432)!( 1, 10)( 2, 12)( 3, 11)( 4, 14)( 5, 13)( 6, 15)( 7, 18)
( 8, 17)( 9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)
( 52, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 68)( 59, 67)( 60, 69)( 61, 72)
( 62, 71)( 63, 70)( 74, 75)( 76, 77)( 79, 81)( 82, 91)( 83, 93)( 84, 92)
( 85, 95)( 86, 94)( 87, 96)( 88, 99)( 89, 98)( 90, 97)(101,102)(103,104)
(106,108)(109,145)(110,147)(111,146)(112,149)(113,148)(114,150)(115,153)
(116,152)(117,151)(118,136)(119,138)(120,137)(121,140)(122,139)(123,141)
(124,144)(125,143)(126,142)(127,154)(128,156)(129,155)(130,158)(131,157)
(132,159)(133,162)(134,161)(135,160)(163,199)(164,201)(165,200)(166,203)
(167,202)(168,204)(169,207)(170,206)(171,205)(172,190)(173,192)(174,191)
(175,194)(176,193)(177,195)(178,198)(179,197)(180,196)(181,208)(182,210)
(183,209)(184,212)(185,211)(186,213)(187,216)(188,215)(189,214)(217,280)
(218,282)(219,281)(220,284)(221,283)(222,285)(223,288)(224,287)(225,286)
(226,271)(227,273)(228,272)(229,275)(230,274)(231,276)(232,279)(233,278)
(234,277)(235,289)(236,291)(237,290)(238,293)(239,292)(240,294)(241,297)
(242,296)(243,295)(244,307)(245,309)(246,308)(247,311)(248,310)(249,312)
(250,315)(251,314)(252,313)(253,298)(254,300)(255,299)(256,302)(257,301)
(258,303)(259,306)(260,305)(261,304)(262,316)(263,318)(264,317)(265,320)
(266,319)(267,321)(268,324)(269,323)(270,322)(325,415)(326,417)(327,416)
(328,419)(329,418)(330,420)(331,423)(332,422)(333,421)(334,406)(335,408)
(336,407)(337,410)(338,409)(339,411)(340,414)(341,413)(342,412)(343,424)
(344,426)(345,425)(346,428)(347,427)(348,429)(349,432)(350,431)(351,430)
(352,388)(353,390)(354,389)(355,392)(356,391)(357,393)(358,396)(359,395)
(360,394)(361,379)(362,381)(363,380)(364,383)(365,382)(366,384)(367,387)
(368,386)(369,385)(370,397)(371,399)(372,398)(373,401)(374,400)(375,402)
(376,405)(377,404)(378,403);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope