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