Polytope of Type {12,72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,72}*1728a
Also Known As : {12,72|2}. if this polytope has another name.
Group : SmallGroup(1728,4714)
Rank : 3
Schlafli Type : {12,72}
Number of vertices, edges, etc : 12, 432, 72
Order of s0s1s2 : 72
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {6,72}*864a, {12,36}*864a
3-fold quotients : {4,72}*576a, {12,24}*576c
4-fold quotients : {6,36}*432a, {12,18}*432a
6-fold quotients : {4,36}*288a, {2,72}*288, {6,24}*288a, {12,12}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {4,24}*192a, {12,8}*192a
12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144a, {12,6}*144a
18-fold quotients : {4,12}*96a, {12,4}*96a, {2,24}*96, {6,8}*96
24-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {4,8}*64a
36-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
48-fold quotients : {2,9}*36
54-fold quotients : {4,4}*32, {2,8}*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 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)(173,182)
(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)(199,208)
(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)(207,216)
(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)(224,278)
(225,279)(226,289)(227,290)(228,291)(229,292)(230,293)(231,294)(232,295)
(233,296)(234,297)(235,280)(236,281)(237,282)(238,283)(239,284)(240,285)
(241,286)(242,287)(243,288)(244,298)(245,299)(246,300)(247,301)(248,302)
(249,303)(250,304)(251,305)(252,306)(253,316)(254,317)(255,318)(256,319)
(257,320)(258,321)(259,322)(260,323)(261,324)(262,307)(263,308)(264,309)
(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(325,379)(326,380)
(327,381)(328,382)(329,383)(330,384)(331,385)(332,386)(333,387)(334,397)
(335,398)(336,399)(337,400)(338,401)(339,402)(340,403)(341,404)(342,405)
(343,388)(344,389)(345,390)(346,391)(347,392)(348,393)(349,394)(350,395)
(351,396)(352,406)(353,407)(354,408)(355,409)(356,410)(357,411)(358,412)
(359,413)(360,414)(361,424)(362,425)(363,426)(364,427)(365,428)(366,429)
(367,430)(368,431)(369,432)(370,415)(371,416)(372,417)(373,418)(374,419)
(375,420)(376,421)(377,422)(378,423);;
s1 := ( 1,226)( 2,228)( 3,227)( 4,234)( 5,233)( 6,232)( 7,231)( 8,230)
( 9,229)( 10,217)( 11,219)( 12,218)( 13,225)( 14,224)( 15,223)( 16,222)
( 17,221)( 18,220)( 19,235)( 20,237)( 21,236)( 22,243)( 23,242)( 24,241)
( 25,240)( 26,239)( 27,238)( 28,253)( 29,255)( 30,254)( 31,261)( 32,260)
( 33,259)( 34,258)( 35,257)( 36,256)( 37,244)( 38,246)( 39,245)( 40,252)
( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,262)( 47,264)( 48,263)
( 49,270)( 50,269)( 51,268)( 52,267)( 53,266)( 54,265)( 55,280)( 56,282)
( 57,281)( 58,288)( 59,287)( 60,286)( 61,285)( 62,284)( 63,283)( 64,271)
( 65,273)( 66,272)( 67,279)( 68,278)( 69,277)( 70,276)( 71,275)( 72,274)
( 73,289)( 74,291)( 75,290)( 76,297)( 77,296)( 78,295)( 79,294)( 80,293)
( 81,292)( 82,307)( 83,309)( 84,308)( 85,315)( 86,314)( 87,313)( 88,312)
( 89,311)( 90,310)( 91,298)( 92,300)( 93,299)( 94,306)( 95,305)( 96,304)
( 97,303)( 98,302)( 99,301)(100,316)(101,318)(102,317)(103,324)(104,323)
(105,322)(106,321)(107,320)(108,319)(109,361)(110,363)(111,362)(112,369)
(113,368)(114,367)(115,366)(116,365)(117,364)(118,352)(119,354)(120,353)
(121,360)(122,359)(123,358)(124,357)(125,356)(126,355)(127,370)(128,372)
(129,371)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)(136,334)
(137,336)(138,335)(139,342)(140,341)(141,340)(142,339)(143,338)(144,337)
(145,325)(146,327)(147,326)(148,333)(149,332)(150,331)(151,330)(152,329)
(153,328)(154,343)(155,345)(156,344)(157,351)(158,350)(159,349)(160,348)
(161,347)(162,346)(163,415)(164,417)(165,416)(166,423)(167,422)(168,421)
(169,420)(170,419)(171,418)(172,406)(173,408)(174,407)(175,414)(176,413)
(177,412)(178,411)(179,410)(180,409)(181,424)(182,426)(183,425)(184,432)
(185,431)(186,430)(187,429)(188,428)(189,427)(190,388)(191,390)(192,389)
(193,396)(194,395)(195,394)(196,393)(197,392)(198,391)(199,379)(200,381)
(201,380)(202,387)(203,386)(204,385)(205,384)(206,383)(207,382)(208,397)
(209,399)(210,398)(211,405)(212,404)(213,403)(214,402)(215,401)(216,400);;
s2 := ( 1, 4)( 2, 6)( 3, 5)( 7, 9)( 10, 13)( 11, 15)( 12, 14)( 16, 18)
( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)( 34, 36)
( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)( 52, 54)
( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)( 70, 72)
( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)( 88, 90)
( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)(106,108)
(109,139)(110,141)(111,140)(112,136)(113,138)(114,137)(115,144)(116,143)
(117,142)(118,148)(119,150)(120,149)(121,145)(122,147)(123,146)(124,153)
(125,152)(126,151)(127,157)(128,159)(129,158)(130,154)(131,156)(132,155)
(133,162)(134,161)(135,160)(163,193)(164,195)(165,194)(166,190)(167,192)
(168,191)(169,198)(170,197)(171,196)(172,202)(173,204)(174,203)(175,199)
(176,201)(177,200)(178,207)(179,206)(180,205)(181,211)(182,213)(183,212)
(184,208)(185,210)(186,209)(187,216)(188,215)(189,214)(217,328)(218,330)
(219,329)(220,325)(221,327)(222,326)(223,333)(224,332)(225,331)(226,337)
(227,339)(228,338)(229,334)(230,336)(231,335)(232,342)(233,341)(234,340)
(235,346)(236,348)(237,347)(238,343)(239,345)(240,344)(241,351)(242,350)
(243,349)(244,355)(245,357)(246,356)(247,352)(248,354)(249,353)(250,360)
(251,359)(252,358)(253,364)(254,366)(255,365)(256,361)(257,363)(258,362)
(259,369)(260,368)(261,367)(262,373)(263,375)(264,374)(265,370)(266,372)
(267,371)(268,378)(269,377)(270,376)(271,382)(272,384)(273,383)(274,379)
(275,381)(276,380)(277,387)(278,386)(279,385)(280,391)(281,393)(282,392)
(283,388)(284,390)(285,389)(286,396)(287,395)(288,394)(289,400)(290,402)
(291,401)(292,397)(293,399)(294,398)(295,405)(296,404)(297,403)(298,409)
(299,411)(300,410)(301,406)(302,408)(303,407)(304,414)(305,413)(306,412)
(307,418)(308,420)(309,419)(310,415)(311,417)(312,416)(313,423)(314,422)
(315,421)(316,427)(317,429)(318,428)(319,424)(320,426)(321,425)(322,432)
(323,431)(324,430);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)(172,181)
(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)(180,189)
(199,208)(200,209)(201,210)(202,211)(203,212)(204,213)(205,214)(206,215)
(207,216)(217,271)(218,272)(219,273)(220,274)(221,275)(222,276)(223,277)
(224,278)(225,279)(226,289)(227,290)(228,291)(229,292)(230,293)(231,294)
(232,295)(233,296)(234,297)(235,280)(236,281)(237,282)(238,283)(239,284)
(240,285)(241,286)(242,287)(243,288)(244,298)(245,299)(246,300)(247,301)
(248,302)(249,303)(250,304)(251,305)(252,306)(253,316)(254,317)(255,318)
(256,319)(257,320)(258,321)(259,322)(260,323)(261,324)(262,307)(263,308)
(264,309)(265,310)(266,311)(267,312)(268,313)(269,314)(270,315)(325,379)
(326,380)(327,381)(328,382)(329,383)(330,384)(331,385)(332,386)(333,387)
(334,397)(335,398)(336,399)(337,400)(338,401)(339,402)(340,403)(341,404)
(342,405)(343,388)(344,389)(345,390)(346,391)(347,392)(348,393)(349,394)
(350,395)(351,396)(352,406)(353,407)(354,408)(355,409)(356,410)(357,411)
(358,412)(359,413)(360,414)(361,424)(362,425)(363,426)(364,427)(365,428)
(366,429)(367,430)(368,431)(369,432)(370,415)(371,416)(372,417)(373,418)
(374,419)(375,420)(376,421)(377,422)(378,423);
s1 := Sym(432)!( 1,226)( 2,228)( 3,227)( 4,234)( 5,233)( 6,232)( 7,231)
( 8,230)( 9,229)( 10,217)( 11,219)( 12,218)( 13,225)( 14,224)( 15,223)
( 16,222)( 17,221)( 18,220)( 19,235)( 20,237)( 21,236)( 22,243)( 23,242)
( 24,241)( 25,240)( 26,239)( 27,238)( 28,253)( 29,255)( 30,254)( 31,261)
( 32,260)( 33,259)( 34,258)( 35,257)( 36,256)( 37,244)( 38,246)( 39,245)
( 40,252)( 41,251)( 42,250)( 43,249)( 44,248)( 45,247)( 46,262)( 47,264)
( 48,263)( 49,270)( 50,269)( 51,268)( 52,267)( 53,266)( 54,265)( 55,280)
( 56,282)( 57,281)( 58,288)( 59,287)( 60,286)( 61,285)( 62,284)( 63,283)
( 64,271)( 65,273)( 66,272)( 67,279)( 68,278)( 69,277)( 70,276)( 71,275)
( 72,274)( 73,289)( 74,291)( 75,290)( 76,297)( 77,296)( 78,295)( 79,294)
( 80,293)( 81,292)( 82,307)( 83,309)( 84,308)( 85,315)( 86,314)( 87,313)
( 88,312)( 89,311)( 90,310)( 91,298)( 92,300)( 93,299)( 94,306)( 95,305)
( 96,304)( 97,303)( 98,302)( 99,301)(100,316)(101,318)(102,317)(103,324)
(104,323)(105,322)(106,321)(107,320)(108,319)(109,361)(110,363)(111,362)
(112,369)(113,368)(114,367)(115,366)(116,365)(117,364)(118,352)(119,354)
(120,353)(121,360)(122,359)(123,358)(124,357)(125,356)(126,355)(127,370)
(128,372)(129,371)(130,378)(131,377)(132,376)(133,375)(134,374)(135,373)
(136,334)(137,336)(138,335)(139,342)(140,341)(141,340)(142,339)(143,338)
(144,337)(145,325)(146,327)(147,326)(148,333)(149,332)(150,331)(151,330)
(152,329)(153,328)(154,343)(155,345)(156,344)(157,351)(158,350)(159,349)
(160,348)(161,347)(162,346)(163,415)(164,417)(165,416)(166,423)(167,422)
(168,421)(169,420)(170,419)(171,418)(172,406)(173,408)(174,407)(175,414)
(176,413)(177,412)(178,411)(179,410)(180,409)(181,424)(182,426)(183,425)
(184,432)(185,431)(186,430)(187,429)(188,428)(189,427)(190,388)(191,390)
(192,389)(193,396)(194,395)(195,394)(196,393)(197,392)(198,391)(199,379)
(200,381)(201,380)(202,387)(203,386)(204,385)(205,384)(206,383)(207,382)
(208,397)(209,399)(210,398)(211,405)(212,404)(213,403)(214,402)(215,401)
(216,400);
s2 := Sym(432)!( 1, 4)( 2, 6)( 3, 5)( 7, 9)( 10, 13)( 11, 15)( 12, 14)
( 16, 18)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)
( 34, 36)( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)
( 52, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)
( 70, 72)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)
( 88, 90)( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)
(106,108)(109,139)(110,141)(111,140)(112,136)(113,138)(114,137)(115,144)
(116,143)(117,142)(118,148)(119,150)(120,149)(121,145)(122,147)(123,146)
(124,153)(125,152)(126,151)(127,157)(128,159)(129,158)(130,154)(131,156)
(132,155)(133,162)(134,161)(135,160)(163,193)(164,195)(165,194)(166,190)
(167,192)(168,191)(169,198)(170,197)(171,196)(172,202)(173,204)(174,203)
(175,199)(176,201)(177,200)(178,207)(179,206)(180,205)(181,211)(182,213)
(183,212)(184,208)(185,210)(186,209)(187,216)(188,215)(189,214)(217,328)
(218,330)(219,329)(220,325)(221,327)(222,326)(223,333)(224,332)(225,331)
(226,337)(227,339)(228,338)(229,334)(230,336)(231,335)(232,342)(233,341)
(234,340)(235,346)(236,348)(237,347)(238,343)(239,345)(240,344)(241,351)
(242,350)(243,349)(244,355)(245,357)(246,356)(247,352)(248,354)(249,353)
(250,360)(251,359)(252,358)(253,364)(254,366)(255,365)(256,361)(257,363)
(258,362)(259,369)(260,368)(261,367)(262,373)(263,375)(264,374)(265,370)
(266,372)(267,371)(268,378)(269,377)(270,376)(271,382)(272,384)(273,383)
(274,379)(275,381)(276,380)(277,387)(278,386)(279,385)(280,391)(281,393)
(282,392)(283,388)(284,390)(285,389)(286,396)(287,395)(288,394)(289,400)
(290,402)(291,401)(292,397)(293,399)(294,398)(295,405)(296,404)(297,403)
(298,409)(299,411)(300,410)(301,406)(302,408)(303,407)(304,414)(305,413)
(306,412)(307,418)(308,420)(309,419)(310,415)(311,417)(312,416)(313,423)
(314,422)(315,421)(316,427)(317,429)(318,428)(319,424)(320,426)(321,425)
(322,432)(323,431)(324,430);
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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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