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