Polytope of Type {12,72}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,72}*1728c
if this polytope has a name.
Group : SmallGroup(1728,5197)
Rank : 3
Schlafli Type : {12,72}
Number of vertices, edges, etc : 12, 432, 72
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
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 : {12,36}*864a
3-fold quotients : {4,72}*576b, {12,24}*576e
4-fold quotients : {6,36}*432a, {12,18}*432a
6-fold quotients : {4,36}*288a, {12,12}*288a
8-fold quotients : {6,18}*216a
9-fold quotients : {4,24}*192b, {12,8}*192b
12-fold quotients : {2,36}*144, {4,18}*144a, {6,12}*144a, {12,6}*144a
18-fold quotients : {4,12}*96a, {12,4}*96a
24-fold quotients : {2,18}*72, {6,6}*72a
27-fold quotients : {4,8}*64b
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
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)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)(120,156)
(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)(128,146)
(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)
(188,206)(189,207)(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)
(325,406)(326,407)(327,408)(328,409)(329,410)(330,411)(331,412)(332,413)
(333,414)(334,424)(335,425)(336,426)(337,427)(338,428)(339,429)(340,430)
(341,431)(342,432)(343,415)(344,416)(345,417)(346,418)(347,419)(348,420)
(349,421)(350,422)(351,423)(352,379)(353,380)(354,381)(355,382)(356,383)
(357,384)(358,385)(359,386)(360,387)(361,397)(362,398)(363,399)(364,400)
(365,401)(366,402)(367,403)(368,404)(369,405)(370,388)(371,389)(372,390)
(373,391)(374,392)(375,393)(376,394)(377,395)(378,396);;
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, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 90)( 62, 89)
( 63, 88)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)( 70, 99)
( 71, 98)( 72, 97)( 73,103)( 74,105)( 75,104)( 76,100)( 77,102)( 78,101)
( 79,108)( 80,107)( 81,106)(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)(163,166)(164,168)
(165,167)(169,171)(172,175)(173,177)(174,176)(178,180)(181,184)(182,186)
(183,185)(187,189)(190,193)(191,195)(192,194)(196,198)(199,202)(200,204)
(201,203)(205,207)(208,211)(209,213)(210,212)(214,216)(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)(271,409)(272,411)(273,410)(274,406)
(275,408)(276,407)(277,414)(278,413)(279,412)(280,418)(281,420)(282,419)
(283,415)(284,417)(285,416)(286,423)(287,422)(288,421)(289,427)(290,429)
(291,428)(292,424)(293,426)(294,425)(295,432)(296,431)(297,430)(298,382)
(299,384)(300,383)(301,379)(302,381)(303,380)(304,387)(305,386)(306,385)
(307,391)(308,393)(309,392)(310,388)(311,390)(312,389)(313,396)(314,395)
(315,394)(316,400)(317,402)(318,401)(319,397)(320,399)(321,398)(322,405)
(323,404)(324,403);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*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,
s2*s0*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*s0*s1*s2*s1 ];;
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)(109,136)(110,137)(111,138)
(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)
(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)
(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207)(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)(325,406)(326,407)(327,408)(328,409)(329,410)(330,411)(331,412)
(332,413)(333,414)(334,424)(335,425)(336,426)(337,427)(338,428)(339,429)
(340,430)(341,431)(342,432)(343,415)(344,416)(345,417)(346,418)(347,419)
(348,420)(349,421)(350,422)(351,423)(352,379)(353,380)(354,381)(355,382)
(356,383)(357,384)(358,385)(359,386)(360,387)(361,397)(362,398)(363,399)
(364,400)(365,401)(366,402)(367,403)(368,404)(369,405)(370,388)(371,389)
(372,390)(373,391)(374,392)(375,393)(376,394)(377,395)(378,396);
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, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 90)
( 62, 89)( 63, 88)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)
( 70, 99)( 71, 98)( 72, 97)( 73,103)( 74,105)( 75,104)( 76,100)( 77,102)
( 78,101)( 79,108)( 80,107)( 81,106)(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)(163,166)
(164,168)(165,167)(169,171)(172,175)(173,177)(174,176)(178,180)(181,184)
(182,186)(183,185)(187,189)(190,193)(191,195)(192,194)(196,198)(199,202)
(200,204)(201,203)(205,207)(208,211)(209,213)(210,212)(214,216)(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)(271,409)(272,411)(273,410)
(274,406)(275,408)(276,407)(277,414)(278,413)(279,412)(280,418)(281,420)
(282,419)(283,415)(284,417)(285,416)(286,423)(287,422)(288,421)(289,427)
(290,429)(291,428)(292,424)(293,426)(294,425)(295,432)(296,431)(297,430)
(298,382)(299,384)(300,383)(301,379)(302,381)(303,380)(304,387)(305,386)
(306,385)(307,391)(308,393)(309,392)(310,388)(311,390)(312,389)(313,396)
(314,395)(315,394)(316,400)(317,402)(318,401)(319,397)(320,399)(321,398)
(322,405)(323,404)(324,403);
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*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,
s2*s0*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*s0*s1*s2*s1 >;
References : None.
to this polytope