Polytope of Type {6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30298)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 72, 432, 144
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,6}*864b
3-fold quotients : {6,12}*576d
4-fold quotients : {6,3}*432
6-fold quotients : {6,6}*288a
8-fold quotients : {6,6}*216a
9-fold quotients : {6,12}*192b
12-fold quotients : {6,3}*144
16-fold quotients : {6,3}*108
18-fold quotients : {3,12}*96, {6,6}*96
24-fold quotients : {6,6}*72b
36-fold quotients : {3,6}*48, {6,3}*48
48-fold quotients : {6,3}*36
72-fold quotients : {3,3}*24, {2,6}*24
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 := ( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)( 79,149)
( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)( 85,167)( 86,168)( 87,165)
( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)( 93,159)( 94,160)( 95,157)
( 96,158)( 97,169)( 98,170)( 99,172)(100,171)(101,175)(102,176)(103,173)
(104,174)(105,185)(106,186)(107,188)(108,187)(109,191)(110,192)(111,189)
(112,190)(113,177)(114,178)(115,180)(116,179)(117,183)(118,184)(119,181)
(120,182)(121,193)(122,194)(123,196)(124,195)(125,199)(126,200)(127,197)
(128,198)(129,209)(130,210)(131,212)(132,211)(133,215)(134,216)(135,213)
(136,214)(137,201)(138,202)(139,204)(140,203)(141,207)(142,208)(143,205)
(144,206)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)(228,236)
(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)(249,258)
(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)(265,266)
(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)(278,287)
(279,286)(280,285)(289,362)(290,361)(291,363)(292,364)(293,368)(294,367)
(295,366)(296,365)(297,378)(298,377)(299,379)(300,380)(301,384)(302,383)
(303,382)(304,381)(305,370)(306,369)(307,371)(308,372)(309,376)(310,375)
(311,374)(312,373)(313,386)(314,385)(315,387)(316,388)(317,392)(318,391)
(319,390)(320,389)(321,402)(322,401)(323,403)(324,404)(325,408)(326,407)
(327,406)(328,405)(329,394)(330,393)(331,395)(332,396)(333,400)(334,399)
(335,398)(336,397)(337,410)(338,409)(339,411)(340,412)(341,416)(342,415)
(343,414)(344,413)(345,426)(346,425)(347,427)(348,428)(349,432)(350,431)
(351,430)(352,429)(353,418)(354,417)(355,419)(356,420)(357,424)(358,423)
(359,422)(360,421);;
s1 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)( 16, 84)
( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)( 24, 92)
( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)( 32,140)
( 33,121)( 34,122)( 35,127)( 36,128)( 37,126)( 38,125)( 39,123)( 40,124)
( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)( 48,132)
( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)( 56,108)
( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)( 64,116)
( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)( 72,100)
(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)(164,168)
(165,166)(169,209)(170,210)(171,215)(172,216)(173,214)(174,213)(175,211)
(176,212)(177,193)(178,194)(179,199)(180,200)(181,198)(182,197)(183,195)
(184,196)(185,201)(186,202)(187,207)(188,208)(189,206)(190,205)(191,203)
(192,204)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)(223,292)
(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)(230,302)(231,300)
(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)(238,310)(239,308)
(240,307)(241,354)(242,353)(243,360)(244,359)(245,357)(246,358)(247,356)
(248,355)(249,338)(250,337)(251,344)(252,343)(253,341)(254,342)(255,340)
(256,339)(257,346)(258,345)(259,352)(260,351)(261,349)(262,350)(263,348)
(264,347)(265,322)(266,321)(267,328)(268,327)(269,325)(270,326)(271,324)
(272,323)(273,330)(274,329)(275,336)(276,335)(277,333)(278,334)(279,332)
(280,331)(281,314)(282,313)(283,320)(284,319)(285,317)(286,318)(287,316)
(288,315)(361,362)(363,368)(364,367)(369,370)(371,376)(372,375)(377,378)
(379,384)(380,383)(385,426)(386,425)(387,432)(388,431)(389,429)(390,430)
(391,428)(392,427)(393,410)(394,409)(395,416)(396,415)(397,413)(398,414)
(399,412)(400,411)(401,418)(402,417)(403,424)(404,423)(405,421)(406,422)
(407,420)(408,419);;
s2 := ( 1,243)( 2,244)( 3,241)( 4,242)( 5,246)( 6,245)( 7,247)( 8,248)
( 9,251)( 10,252)( 11,249)( 12,250)( 13,254)( 14,253)( 15,255)( 16,256)
( 17,259)( 18,260)( 19,257)( 20,258)( 21,262)( 22,261)( 23,263)( 24,264)
( 25,219)( 26,220)( 27,217)( 28,218)( 29,222)( 30,221)( 31,223)( 32,224)
( 33,227)( 34,228)( 35,225)( 36,226)( 37,230)( 38,229)( 39,231)( 40,232)
( 41,235)( 42,236)( 43,233)( 44,234)( 45,238)( 46,237)( 47,239)( 48,240)
( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)( 56,272)
( 57,275)( 58,276)( 59,273)( 60,274)( 61,278)( 62,277)( 63,279)( 64,280)
( 65,283)( 66,284)( 67,281)( 68,282)( 69,286)( 70,285)( 71,287)( 72,288)
( 73,387)( 74,388)( 75,385)( 76,386)( 77,390)( 78,389)( 79,391)( 80,392)
( 81,395)( 82,396)( 83,393)( 84,394)( 85,398)( 86,397)( 87,399)( 88,400)
( 89,403)( 90,404)( 91,401)( 92,402)( 93,406)( 94,405)( 95,407)( 96,408)
( 97,363)( 98,364)( 99,361)(100,362)(101,366)(102,365)(103,367)(104,368)
(105,371)(106,372)(107,369)(108,370)(109,374)(110,373)(111,375)(112,376)
(113,379)(114,380)(115,377)(116,378)(117,382)(118,381)(119,383)(120,384)
(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)(128,416)
(129,419)(130,420)(131,417)(132,418)(133,422)(134,421)(135,423)(136,424)
(137,427)(138,428)(139,425)(140,426)(141,430)(142,429)(143,431)(144,432)
(145,315)(146,316)(147,313)(148,314)(149,318)(150,317)(151,319)(152,320)
(153,323)(154,324)(155,321)(156,322)(157,326)(158,325)(159,327)(160,328)
(161,331)(162,332)(163,329)(164,330)(165,334)(166,333)(167,335)(168,336)
(169,291)(170,292)(171,289)(172,290)(173,294)(174,293)(175,295)(176,296)
(177,299)(178,300)(179,297)(180,298)(181,302)(182,301)(183,303)(184,304)
(185,307)(186,308)(187,305)(188,306)(189,310)(190,309)(191,311)(192,312)
(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)(200,344)
(201,347)(202,348)(203,345)(204,346)(205,350)(206,349)(207,351)(208,352)
(209,355)(210,356)(211,353)(212,354)(213,358)(214,357)(215,359)(216,360);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)
( 79,149)( 80,150)( 81,161)( 82,162)( 83,164)( 84,163)( 85,167)( 86,168)
( 87,165)( 88,166)( 89,153)( 90,154)( 91,156)( 92,155)( 93,159)( 94,160)
( 95,157)( 96,158)( 97,169)( 98,170)( 99,172)(100,171)(101,175)(102,176)
(103,173)(104,174)(105,185)(106,186)(107,188)(108,187)(109,191)(110,192)
(111,189)(112,190)(113,177)(114,178)(115,180)(116,179)(117,183)(118,184)
(119,181)(120,182)(121,193)(122,194)(123,196)(124,195)(125,199)(126,200)
(127,197)(128,198)(129,209)(130,210)(131,212)(132,211)(133,215)(134,216)
(135,213)(136,214)(137,201)(138,202)(139,204)(140,203)(141,207)(142,208)
(143,205)(144,206)(217,218)(221,224)(222,223)(225,234)(226,233)(227,235)
(228,236)(229,240)(230,239)(231,238)(232,237)(241,242)(245,248)(246,247)
(249,258)(250,257)(251,259)(252,260)(253,264)(254,263)(255,262)(256,261)
(265,266)(269,272)(270,271)(273,282)(274,281)(275,283)(276,284)(277,288)
(278,287)(279,286)(280,285)(289,362)(290,361)(291,363)(292,364)(293,368)
(294,367)(295,366)(296,365)(297,378)(298,377)(299,379)(300,380)(301,384)
(302,383)(303,382)(304,381)(305,370)(306,369)(307,371)(308,372)(309,376)
(310,375)(311,374)(312,373)(313,386)(314,385)(315,387)(316,388)(317,392)
(318,391)(319,390)(320,389)(321,402)(322,401)(323,403)(324,404)(325,408)
(326,407)(327,406)(328,405)(329,394)(330,393)(331,395)(332,396)(333,400)
(334,399)(335,398)(336,397)(337,410)(338,409)(339,411)(340,412)(341,416)
(342,415)(343,414)(344,413)(345,426)(346,425)(347,427)(348,428)(349,432)
(350,431)(351,430)(352,429)(353,418)(354,417)(355,419)(356,420)(357,424)
(358,423)(359,422)(360,421);
s1 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 81)( 10, 82)( 11, 87)( 12, 88)( 13, 86)( 14, 85)( 15, 83)
( 16, 84)( 17, 89)( 18, 90)( 19, 95)( 20, 96)( 21, 94)( 22, 93)( 23, 91)
( 24, 92)( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)
( 32,140)( 33,121)( 34,122)( 35,127)( 36,128)( 37,126)( 38,125)( 39,123)
( 40,124)( 41,129)( 42,130)( 43,135)( 44,136)( 45,134)( 46,133)( 47,131)
( 48,132)( 49,105)( 50,106)( 51,111)( 52,112)( 53,110)( 54,109)( 55,107)
( 56,108)( 57,113)( 58,114)( 59,119)( 60,120)( 61,118)( 62,117)( 63,115)
( 64,116)( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)
( 72,100)(147,151)(148,152)(149,150)(155,159)(156,160)(157,158)(163,167)
(164,168)(165,166)(169,209)(170,210)(171,215)(172,216)(173,214)(174,213)
(175,211)(176,212)(177,193)(178,194)(179,199)(180,200)(181,198)(182,197)
(183,195)(184,196)(185,201)(186,202)(187,207)(188,208)(189,206)(190,205)
(191,203)(192,204)(217,290)(218,289)(219,296)(220,295)(221,293)(222,294)
(223,292)(224,291)(225,298)(226,297)(227,304)(228,303)(229,301)(230,302)
(231,300)(232,299)(233,306)(234,305)(235,312)(236,311)(237,309)(238,310)
(239,308)(240,307)(241,354)(242,353)(243,360)(244,359)(245,357)(246,358)
(247,356)(248,355)(249,338)(250,337)(251,344)(252,343)(253,341)(254,342)
(255,340)(256,339)(257,346)(258,345)(259,352)(260,351)(261,349)(262,350)
(263,348)(264,347)(265,322)(266,321)(267,328)(268,327)(269,325)(270,326)
(271,324)(272,323)(273,330)(274,329)(275,336)(276,335)(277,333)(278,334)
(279,332)(280,331)(281,314)(282,313)(283,320)(284,319)(285,317)(286,318)
(287,316)(288,315)(361,362)(363,368)(364,367)(369,370)(371,376)(372,375)
(377,378)(379,384)(380,383)(385,426)(386,425)(387,432)(388,431)(389,429)
(390,430)(391,428)(392,427)(393,410)(394,409)(395,416)(396,415)(397,413)
(398,414)(399,412)(400,411)(401,418)(402,417)(403,424)(404,423)(405,421)
(406,422)(407,420)(408,419);
s2 := Sym(432)!( 1,243)( 2,244)( 3,241)( 4,242)( 5,246)( 6,245)( 7,247)
( 8,248)( 9,251)( 10,252)( 11,249)( 12,250)( 13,254)( 14,253)( 15,255)
( 16,256)( 17,259)( 18,260)( 19,257)( 20,258)( 21,262)( 22,261)( 23,263)
( 24,264)( 25,219)( 26,220)( 27,217)( 28,218)( 29,222)( 30,221)( 31,223)
( 32,224)( 33,227)( 34,228)( 35,225)( 36,226)( 37,230)( 38,229)( 39,231)
( 40,232)( 41,235)( 42,236)( 43,233)( 44,234)( 45,238)( 46,237)( 47,239)
( 48,240)( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)
( 56,272)( 57,275)( 58,276)( 59,273)( 60,274)( 61,278)( 62,277)( 63,279)
( 64,280)( 65,283)( 66,284)( 67,281)( 68,282)( 69,286)( 70,285)( 71,287)
( 72,288)( 73,387)( 74,388)( 75,385)( 76,386)( 77,390)( 78,389)( 79,391)
( 80,392)( 81,395)( 82,396)( 83,393)( 84,394)( 85,398)( 86,397)( 87,399)
( 88,400)( 89,403)( 90,404)( 91,401)( 92,402)( 93,406)( 94,405)( 95,407)
( 96,408)( 97,363)( 98,364)( 99,361)(100,362)(101,366)(102,365)(103,367)
(104,368)(105,371)(106,372)(107,369)(108,370)(109,374)(110,373)(111,375)
(112,376)(113,379)(114,380)(115,377)(116,378)(117,382)(118,381)(119,383)
(120,384)(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)
(128,416)(129,419)(130,420)(131,417)(132,418)(133,422)(134,421)(135,423)
(136,424)(137,427)(138,428)(139,425)(140,426)(141,430)(142,429)(143,431)
(144,432)(145,315)(146,316)(147,313)(148,314)(149,318)(150,317)(151,319)
(152,320)(153,323)(154,324)(155,321)(156,322)(157,326)(158,325)(159,327)
(160,328)(161,331)(162,332)(163,329)(164,330)(165,334)(166,333)(167,335)
(168,336)(169,291)(170,292)(171,289)(172,290)(173,294)(174,293)(175,295)
(176,296)(177,299)(178,300)(179,297)(180,298)(181,302)(182,301)(183,303)
(184,304)(185,307)(186,308)(187,305)(188,306)(189,310)(190,309)(191,311)
(192,312)(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)
(200,344)(201,347)(202,348)(203,345)(204,346)(205,350)(206,349)(207,351)
(208,352)(209,355)(210,356)(211,353)(212,354)(213,358)(214,357)(215,359)
(216,360);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope