Polytope of Type {6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,30272)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 144, 432, 144
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,6}*864a
3-fold quotients : {6,6}*576b
4-fold quotients : {3,6}*432
6-fold quotients : {6,6}*288b
8-fold quotients : {6,6}*216c
9-fold quotients : {6,6}*192b
12-fold quotients : {3,6}*144
16-fold quotients : {3,6}*108
18-fold quotients : {6,6}*96
24-fold quotients : {6,6}*72c
36-fold quotients : {3,6}*48, {6,3}*48
48-fold quotients : {3,6}*36
72-fold quotients : {3,3}*24, {6,2}*24
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 := ( 1,217)( 2,218)( 3,220)( 4,219)( 5,223)( 6,224)( 7,221)( 8,222)
( 9,225)( 10,226)( 11,228)( 12,227)( 13,231)( 14,232)( 15,229)( 16,230)
( 17,233)( 18,234)( 19,236)( 20,235)( 21,239)( 22,240)( 23,237)( 24,238)
( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)( 32,270)
( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)( 40,278)
( 41,281)( 42,282)( 43,284)( 44,283)( 45,287)( 46,288)( 47,285)( 48,286)
( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)( 56,246)
( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)( 64,254)
( 65,257)( 66,258)( 67,260)( 68,259)( 69,263)( 70,264)( 71,261)( 72,262)
( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)( 80,366)
( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)( 88,374)
( 89,377)( 90,378)( 91,380)( 92,379)( 93,383)( 94,384)( 95,381)( 96,382)
( 97,409)( 98,410)( 99,412)(100,411)(101,415)(102,416)(103,413)(104,414)
(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)(112,422)
(113,425)(114,426)(115,428)(116,427)(117,431)(118,432)(119,429)(120,430)
(121,385)(122,386)(123,388)(124,387)(125,391)(126,392)(127,389)(128,390)
(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)(136,398)
(137,401)(138,402)(139,404)(140,403)(141,407)(142,408)(143,405)(144,406)
(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)(152,294)
(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)(160,302)
(161,305)(162,306)(163,308)(164,307)(165,311)(166,312)(167,309)(168,310)
(169,337)(170,338)(171,340)(172,339)(173,343)(174,344)(175,341)(176,342)
(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)(184,350)
(185,353)(186,354)(187,356)(188,355)(189,359)(190,360)(191,357)(192,358)
(193,313)(194,314)(195,316)(196,315)(197,319)(198,320)(199,317)(200,318)
(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)(208,326)
(209,329)(210,330)(211,332)(212,331)(213,335)(214,336)(215,333)(216,334);;
s1 := ( 1,177)( 2,178)( 3,182)( 4,181)( 5,180)( 6,179)( 7,184)( 8,183)
( 9,185)( 10,186)( 11,190)( 12,189)( 13,188)( 14,187)( 15,192)( 16,191)
( 17,169)( 18,170)( 19,174)( 20,173)( 21,172)( 22,171)( 23,176)( 24,175)
( 25,161)( 26,162)( 27,166)( 28,165)( 29,164)( 30,163)( 31,168)( 32,167)
( 33,145)( 34,146)( 35,150)( 36,149)( 37,148)( 38,147)( 39,152)( 40,151)
( 41,153)( 42,154)( 43,158)( 44,157)( 45,156)( 46,155)( 47,160)( 48,159)
( 49,193)( 50,194)( 51,198)( 52,197)( 53,196)( 54,195)( 55,200)( 56,199)
( 57,201)( 58,202)( 59,206)( 60,205)( 61,204)( 62,203)( 63,208)( 64,207)
( 65,209)( 66,210)( 67,214)( 68,213)( 69,212)( 70,211)( 71,216)( 72,215)
( 73,105)( 74,106)( 75,110)( 76,109)( 77,108)( 78,107)( 79,112)( 80,111)
( 81,113)( 82,114)( 83,118)( 84,117)( 85,116)( 86,115)( 87,120)( 88,119)
( 89, 97)( 90, 98)( 91,102)( 92,101)( 93,100)( 94, 99)( 95,104)( 96,103)
(123,126)(124,125)(127,128)(131,134)(132,133)(135,136)(139,142)(140,141)
(143,144)(217,393)(218,394)(219,398)(220,397)(221,396)(222,395)(223,400)
(224,399)(225,401)(226,402)(227,406)(228,405)(229,404)(230,403)(231,408)
(232,407)(233,385)(234,386)(235,390)(236,389)(237,388)(238,387)(239,392)
(240,391)(241,377)(242,378)(243,382)(244,381)(245,380)(246,379)(247,384)
(248,383)(249,361)(250,362)(251,366)(252,365)(253,364)(254,363)(255,368)
(256,367)(257,369)(258,370)(259,374)(260,373)(261,372)(262,371)(263,376)
(264,375)(265,409)(266,410)(267,414)(268,413)(269,412)(270,411)(271,416)
(272,415)(273,417)(274,418)(275,422)(276,421)(277,420)(278,419)(279,424)
(280,423)(281,425)(282,426)(283,430)(284,429)(285,428)(286,427)(287,432)
(288,431)(289,321)(290,322)(291,326)(292,325)(293,324)(294,323)(295,328)
(296,327)(297,329)(298,330)(299,334)(300,333)(301,332)(302,331)(303,336)
(304,335)(305,313)(306,314)(307,318)(308,317)(309,316)(310,315)(311,320)
(312,319)(339,342)(340,341)(343,344)(347,350)(348,349)(351,352)(355,358)
(356,357)(359,360);;
s2 := ( 1, 3)( 2, 4)( 5, 6)( 9, 19)( 10, 20)( 11, 17)( 12, 18)( 13, 22)
( 14, 21)( 15, 23)( 16, 24)( 25, 27)( 26, 28)( 29, 30)( 33, 43)( 34, 44)
( 35, 41)( 36, 42)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 49, 51)( 50, 52)
( 53, 54)( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 61, 70)( 62, 69)( 63, 71)
( 64, 72)( 73,147)( 74,148)( 75,145)( 76,146)( 77,150)( 78,149)( 79,151)
( 80,152)( 81,163)( 82,164)( 83,161)( 84,162)( 85,166)( 86,165)( 87,167)
( 88,168)( 89,155)( 90,156)( 91,153)( 92,154)( 93,158)( 94,157)( 95,159)
( 96,160)( 97,171)( 98,172)( 99,169)(100,170)(101,174)(102,173)(103,175)
(104,176)(105,187)(106,188)(107,185)(108,186)(109,190)(110,189)(111,191)
(112,192)(113,179)(114,180)(115,177)(116,178)(117,182)(118,181)(119,183)
(120,184)(121,195)(122,196)(123,193)(124,194)(125,198)(126,197)(127,199)
(128,200)(129,211)(130,212)(131,209)(132,210)(133,214)(134,213)(135,215)
(136,216)(137,203)(138,204)(139,201)(140,202)(141,206)(142,205)(143,207)
(144,208)(217,220)(218,219)(223,224)(225,236)(226,235)(227,234)(228,233)
(229,237)(230,238)(231,240)(232,239)(241,244)(242,243)(247,248)(249,260)
(250,259)(251,258)(252,257)(253,261)(254,262)(255,264)(256,263)(265,268)
(266,267)(271,272)(273,284)(274,283)(275,282)(276,281)(277,285)(278,286)
(279,288)(280,287)(289,364)(290,363)(291,362)(292,361)(293,365)(294,366)
(295,368)(296,367)(297,380)(298,379)(299,378)(300,377)(301,381)(302,382)
(303,384)(304,383)(305,372)(306,371)(307,370)(308,369)(309,373)(310,374)
(311,376)(312,375)(313,388)(314,387)(315,386)(316,385)(317,389)(318,390)
(319,392)(320,391)(321,404)(322,403)(323,402)(324,401)(325,405)(326,406)
(327,408)(328,407)(329,396)(330,395)(331,394)(332,393)(333,397)(334,398)
(335,400)(336,399)(337,412)(338,411)(339,410)(340,409)(341,413)(342,414)
(343,416)(344,415)(345,428)(346,427)(347,426)(348,425)(349,429)(350,430)
(351,432)(352,431)(353,420)(354,419)(355,418)(356,417)(357,421)(358,422)
(359,424)(360,423);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1,217)( 2,218)( 3,220)( 4,219)( 5,223)( 6,224)( 7,221)
( 8,222)( 9,225)( 10,226)( 11,228)( 12,227)( 13,231)( 14,232)( 15,229)
( 16,230)( 17,233)( 18,234)( 19,236)( 20,235)( 21,239)( 22,240)( 23,237)
( 24,238)( 25,265)( 26,266)( 27,268)( 28,267)( 29,271)( 30,272)( 31,269)
( 32,270)( 33,273)( 34,274)( 35,276)( 36,275)( 37,279)( 38,280)( 39,277)
( 40,278)( 41,281)( 42,282)( 43,284)( 44,283)( 45,287)( 46,288)( 47,285)
( 48,286)( 49,241)( 50,242)( 51,244)( 52,243)( 53,247)( 54,248)( 55,245)
( 56,246)( 57,249)( 58,250)( 59,252)( 60,251)( 61,255)( 62,256)( 63,253)
( 64,254)( 65,257)( 66,258)( 67,260)( 68,259)( 69,263)( 70,264)( 71,261)
( 72,262)( 73,361)( 74,362)( 75,364)( 76,363)( 77,367)( 78,368)( 79,365)
( 80,366)( 81,369)( 82,370)( 83,372)( 84,371)( 85,375)( 86,376)( 87,373)
( 88,374)( 89,377)( 90,378)( 91,380)( 92,379)( 93,383)( 94,384)( 95,381)
( 96,382)( 97,409)( 98,410)( 99,412)(100,411)(101,415)(102,416)(103,413)
(104,414)(105,417)(106,418)(107,420)(108,419)(109,423)(110,424)(111,421)
(112,422)(113,425)(114,426)(115,428)(116,427)(117,431)(118,432)(119,429)
(120,430)(121,385)(122,386)(123,388)(124,387)(125,391)(126,392)(127,389)
(128,390)(129,393)(130,394)(131,396)(132,395)(133,399)(134,400)(135,397)
(136,398)(137,401)(138,402)(139,404)(140,403)(141,407)(142,408)(143,405)
(144,406)(145,289)(146,290)(147,292)(148,291)(149,295)(150,296)(151,293)
(152,294)(153,297)(154,298)(155,300)(156,299)(157,303)(158,304)(159,301)
(160,302)(161,305)(162,306)(163,308)(164,307)(165,311)(166,312)(167,309)
(168,310)(169,337)(170,338)(171,340)(172,339)(173,343)(174,344)(175,341)
(176,342)(177,345)(178,346)(179,348)(180,347)(181,351)(182,352)(183,349)
(184,350)(185,353)(186,354)(187,356)(188,355)(189,359)(190,360)(191,357)
(192,358)(193,313)(194,314)(195,316)(196,315)(197,319)(198,320)(199,317)
(200,318)(201,321)(202,322)(203,324)(204,323)(205,327)(206,328)(207,325)
(208,326)(209,329)(210,330)(211,332)(212,331)(213,335)(214,336)(215,333)
(216,334);
s1 := Sym(432)!( 1,177)( 2,178)( 3,182)( 4,181)( 5,180)( 6,179)( 7,184)
( 8,183)( 9,185)( 10,186)( 11,190)( 12,189)( 13,188)( 14,187)( 15,192)
( 16,191)( 17,169)( 18,170)( 19,174)( 20,173)( 21,172)( 22,171)( 23,176)
( 24,175)( 25,161)( 26,162)( 27,166)( 28,165)( 29,164)( 30,163)( 31,168)
( 32,167)( 33,145)( 34,146)( 35,150)( 36,149)( 37,148)( 38,147)( 39,152)
( 40,151)( 41,153)( 42,154)( 43,158)( 44,157)( 45,156)( 46,155)( 47,160)
( 48,159)( 49,193)( 50,194)( 51,198)( 52,197)( 53,196)( 54,195)( 55,200)
( 56,199)( 57,201)( 58,202)( 59,206)( 60,205)( 61,204)( 62,203)( 63,208)
( 64,207)( 65,209)( 66,210)( 67,214)( 68,213)( 69,212)( 70,211)( 71,216)
( 72,215)( 73,105)( 74,106)( 75,110)( 76,109)( 77,108)( 78,107)( 79,112)
( 80,111)( 81,113)( 82,114)( 83,118)( 84,117)( 85,116)( 86,115)( 87,120)
( 88,119)( 89, 97)( 90, 98)( 91,102)( 92,101)( 93,100)( 94, 99)( 95,104)
( 96,103)(123,126)(124,125)(127,128)(131,134)(132,133)(135,136)(139,142)
(140,141)(143,144)(217,393)(218,394)(219,398)(220,397)(221,396)(222,395)
(223,400)(224,399)(225,401)(226,402)(227,406)(228,405)(229,404)(230,403)
(231,408)(232,407)(233,385)(234,386)(235,390)(236,389)(237,388)(238,387)
(239,392)(240,391)(241,377)(242,378)(243,382)(244,381)(245,380)(246,379)
(247,384)(248,383)(249,361)(250,362)(251,366)(252,365)(253,364)(254,363)
(255,368)(256,367)(257,369)(258,370)(259,374)(260,373)(261,372)(262,371)
(263,376)(264,375)(265,409)(266,410)(267,414)(268,413)(269,412)(270,411)
(271,416)(272,415)(273,417)(274,418)(275,422)(276,421)(277,420)(278,419)
(279,424)(280,423)(281,425)(282,426)(283,430)(284,429)(285,428)(286,427)
(287,432)(288,431)(289,321)(290,322)(291,326)(292,325)(293,324)(294,323)
(295,328)(296,327)(297,329)(298,330)(299,334)(300,333)(301,332)(302,331)
(303,336)(304,335)(305,313)(306,314)(307,318)(308,317)(309,316)(310,315)
(311,320)(312,319)(339,342)(340,341)(343,344)(347,350)(348,349)(351,352)
(355,358)(356,357)(359,360);
s2 := Sym(432)!( 1, 3)( 2, 4)( 5, 6)( 9, 19)( 10, 20)( 11, 17)( 12, 18)
( 13, 22)( 14, 21)( 15, 23)( 16, 24)( 25, 27)( 26, 28)( 29, 30)( 33, 43)
( 34, 44)( 35, 41)( 36, 42)( 37, 46)( 38, 45)( 39, 47)( 40, 48)( 49, 51)
( 50, 52)( 53, 54)( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 61, 70)( 62, 69)
( 63, 71)( 64, 72)( 73,147)( 74,148)( 75,145)( 76,146)( 77,150)( 78,149)
( 79,151)( 80,152)( 81,163)( 82,164)( 83,161)( 84,162)( 85,166)( 86,165)
( 87,167)( 88,168)( 89,155)( 90,156)( 91,153)( 92,154)( 93,158)( 94,157)
( 95,159)( 96,160)( 97,171)( 98,172)( 99,169)(100,170)(101,174)(102,173)
(103,175)(104,176)(105,187)(106,188)(107,185)(108,186)(109,190)(110,189)
(111,191)(112,192)(113,179)(114,180)(115,177)(116,178)(117,182)(118,181)
(119,183)(120,184)(121,195)(122,196)(123,193)(124,194)(125,198)(126,197)
(127,199)(128,200)(129,211)(130,212)(131,209)(132,210)(133,214)(134,213)
(135,215)(136,216)(137,203)(138,204)(139,201)(140,202)(141,206)(142,205)
(143,207)(144,208)(217,220)(218,219)(223,224)(225,236)(226,235)(227,234)
(228,233)(229,237)(230,238)(231,240)(232,239)(241,244)(242,243)(247,248)
(249,260)(250,259)(251,258)(252,257)(253,261)(254,262)(255,264)(256,263)
(265,268)(266,267)(271,272)(273,284)(274,283)(275,282)(276,281)(277,285)
(278,286)(279,288)(280,287)(289,364)(290,363)(291,362)(292,361)(293,365)
(294,366)(295,368)(296,367)(297,380)(298,379)(299,378)(300,377)(301,381)
(302,382)(303,384)(304,383)(305,372)(306,371)(307,370)(308,369)(309,373)
(310,374)(311,376)(312,375)(313,388)(314,387)(315,386)(316,385)(317,389)
(318,390)(319,392)(320,391)(321,404)(322,403)(323,402)(324,401)(325,405)
(326,406)(327,408)(328,407)(329,396)(330,395)(331,394)(332,393)(333,397)
(334,398)(335,400)(336,399)(337,412)(338,411)(339,410)(340,409)(341,413)
(342,414)(343,416)(344,415)(345,428)(346,427)(347,426)(348,425)(349,429)
(350,430)(351,432)(352,431)(353,420)(354,419)(355,418)(356,417)(357,421)
(358,422)(359,424)(360,423);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0 >;
References : None.
to this polytope