Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1728d
if this polytope has a name.
Group : SmallGroup(1728,30298)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 144, 432, 72
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 : {12,6}*576d
   4-fold quotients : {3,6}*432
   6-fold quotients : {6,6}*288b
   8-fold quotients : {6,6}*216c
   9-fold quotients : {12,6}*192b
   12-fold quotients : {3,6}*144
   16-fold quotients : {3,6}*108
   18-fold quotients : {12,3}*96, {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 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)( 21, 23)
( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)( 31, 53)
( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)( 39, 61)
( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)( 47, 69)
( 48, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)( 79,149)
( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)( 87,157)
( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)( 93,167)( 94,168)( 95,165)
( 96,166)( 97,193)( 98,194)( 99,196)(100,195)(101,199)(102,200)(103,197)
(104,198)(105,201)(106,202)(107,204)(108,203)(109,207)(110,208)(111,205)
(112,206)(113,209)(114,210)(115,212)(116,211)(117,215)(118,216)(119,213)
(120,214)(121,169)(122,170)(123,172)(124,171)(125,175)(126,176)(127,173)
(128,174)(129,177)(130,178)(131,180)(132,179)(133,183)(134,184)(135,181)
(136,182)(137,185)(138,186)(139,188)(140,187)(141,191)(142,192)(143,189)
(144,190)(217,218)(221,224)(222,223)(225,226)(229,232)(230,231)(233,234)
(237,240)(238,239)(241,266)(242,265)(243,267)(244,268)(245,272)(246,271)
(247,270)(248,269)(249,274)(250,273)(251,275)(252,276)(253,280)(254,279)
(255,278)(256,277)(257,282)(258,281)(259,283)(260,284)(261,288)(262,287)
(263,286)(264,285)(289,362)(290,361)(291,363)(292,364)(293,368)(294,367)
(295,366)(296,365)(297,370)(298,369)(299,371)(300,372)(301,376)(302,375)
(303,374)(304,373)(305,378)(306,377)(307,379)(308,380)(309,384)(310,383)
(311,382)(312,381)(313,410)(314,409)(315,411)(316,412)(317,416)(318,415)
(319,414)(320,413)(321,418)(322,417)(323,419)(324,420)(325,424)(326,423)
(327,422)(328,421)(329,426)(330,425)(331,427)(332,428)(333,432)(334,431)
(335,430)(336,429)(337,386)(338,385)(339,387)(340,388)(341,392)(342,391)
(343,390)(344,389)(345,394)(346,393)(347,395)(348,396)(349,400)(350,399)
(351,398)(352,397)(353,402)(354,401)(355,403)(356,404)(357,408)(358,407)
(359,406)(360,405);;
s1 := (  1,393)(  2,394)(  3,398)(  4,397)(  5,396)(  6,395)(  7,400)(  8,399)
(  9,401)( 10,402)( 11,406)( 12,405)( 13,404)( 14,403)( 15,408)( 16,407)
( 17,385)( 18,386)( 19,390)( 20,389)( 21,388)( 22,387)( 23,392)( 24,391)
( 25,377)( 26,378)( 27,382)( 28,381)( 29,380)( 30,379)( 31,384)( 32,383)
( 33,361)( 34,362)( 35,366)( 36,365)( 37,364)( 38,363)( 39,368)( 40,367)
( 41,369)( 42,370)( 43,374)( 44,373)( 45,372)( 46,371)( 47,376)( 48,375)
( 49,409)( 50,410)( 51,414)( 52,413)( 53,412)( 54,411)( 55,416)( 56,415)
( 57,417)( 58,418)( 59,422)( 60,421)( 61,420)( 62,419)( 63,424)( 64,423)
( 65,425)( 66,426)( 67,430)( 68,429)( 69,428)( 70,427)( 71,432)( 72,431)
( 73,321)( 74,322)( 75,326)( 76,325)( 77,324)( 78,323)( 79,328)( 80,327)
( 81,329)( 82,330)( 83,334)( 84,333)( 85,332)( 86,331)( 87,336)( 88,335)
( 89,313)( 90,314)( 91,318)( 92,317)( 93,316)( 94,315)( 95,320)( 96,319)
( 97,305)( 98,306)( 99,310)(100,309)(101,308)(102,307)(103,312)(104,311)
(105,289)(106,290)(107,294)(108,293)(109,292)(110,291)(111,296)(112,295)
(113,297)(114,298)(115,302)(116,301)(117,300)(118,299)(119,304)(120,303)
(121,337)(122,338)(123,342)(124,341)(125,340)(126,339)(127,344)(128,343)
(129,345)(130,346)(131,350)(132,349)(133,348)(134,347)(135,352)(136,351)
(137,353)(138,354)(139,358)(140,357)(141,356)(142,355)(143,360)(144,359)
(145,249)(146,250)(147,254)(148,253)(149,252)(150,251)(151,256)(152,255)
(153,257)(154,258)(155,262)(156,261)(157,260)(158,259)(159,264)(160,263)
(161,241)(162,242)(163,246)(164,245)(165,244)(166,243)(167,248)(168,247)
(169,233)(170,234)(171,238)(172,237)(173,236)(174,235)(175,240)(176,239)
(177,217)(178,218)(179,222)(180,221)(181,220)(182,219)(183,224)(184,223)
(185,225)(186,226)(187,230)(188,229)(189,228)(190,227)(191,232)(192,231)
(193,265)(194,266)(195,270)(196,269)(197,268)(198,267)(199,272)(200,271)
(201,273)(202,274)(203,278)(204,277)(205,276)(206,275)(207,280)(208,279)
(209,281)(210,282)(211,286)(212,285)(213,284)(214,283)(215,288)(216,287);;
s2 := (  1,219)(  2,220)(  3,217)(  4,218)(  5,222)(  6,221)(  7,223)(  8,224)
(  9,235)( 10,236)( 11,233)( 12,234)( 13,238)( 14,237)( 15,239)( 16,240)
( 17,227)( 18,228)( 19,225)( 20,226)( 21,230)( 22,229)( 23,231)( 24,232)
( 25,243)( 26,244)( 27,241)( 28,242)( 29,246)( 30,245)( 31,247)( 32,248)
( 33,259)( 34,260)( 35,257)( 36,258)( 37,262)( 38,261)( 39,263)( 40,264)
( 41,251)( 42,252)( 43,249)( 44,250)( 45,254)( 46,253)( 47,255)( 48,256)
( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)( 56,272)
( 57,283)( 58,284)( 59,281)( 60,282)( 61,286)( 62,285)( 63,287)( 64,288)
( 65,275)( 66,276)( 67,273)( 68,274)( 69,278)( 70,277)( 71,279)( 72,280)
( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)( 80,368)
( 81,379)( 82,380)( 83,377)( 84,378)( 85,382)( 86,381)( 87,383)( 88,384)
( 89,371)( 90,372)( 91,369)( 92,370)( 93,374)( 94,373)( 95,375)( 96,376)
( 97,387)( 98,388)( 99,385)(100,386)(101,390)(102,389)(103,391)(104,392)
(105,403)(106,404)(107,401)(108,402)(109,406)(110,405)(111,407)(112,408)
(113,395)(114,396)(115,393)(116,394)(117,398)(118,397)(119,399)(120,400)
(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)(128,416)
(129,427)(130,428)(131,425)(132,426)(133,430)(134,429)(135,431)(136,432)
(137,419)(138,420)(139,417)(140,418)(141,422)(142,421)(143,423)(144,424)
(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)(152,296)
(153,307)(154,308)(155,305)(156,306)(157,310)(158,309)(159,311)(160,312)
(161,299)(162,300)(163,297)(164,298)(165,302)(166,301)(167,303)(168,304)
(169,315)(170,316)(171,313)(172,314)(173,318)(174,317)(175,319)(176,320)
(177,331)(178,332)(179,329)(180,330)(181,334)(182,333)(183,335)(184,336)
(185,323)(186,324)(187,321)(188,322)(189,326)(190,325)(191,327)(192,328)
(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)(200,344)
(201,355)(202,356)(203,353)(204,354)(205,358)(206,357)(207,359)(208,360)
(209,347)(210,348)(211,345)(212,346)(213,350)(214,349)(215,351)(216,352);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(432)!(  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 19, 20)
( 21, 23)( 22, 24)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 55)( 30, 56)
( 31, 53)( 32, 54)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)( 38, 64)
( 39, 61)( 40, 62)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 71)( 46, 72)
( 47, 69)( 48, 70)( 73,145)( 74,146)( 75,148)( 76,147)( 77,151)( 78,152)
( 79,149)( 80,150)( 81,153)( 82,154)( 83,156)( 84,155)( 85,159)( 86,160)
( 87,157)( 88,158)( 89,161)( 90,162)( 91,164)( 92,163)( 93,167)( 94,168)
( 95,165)( 96,166)( 97,193)( 98,194)( 99,196)(100,195)(101,199)(102,200)
(103,197)(104,198)(105,201)(106,202)(107,204)(108,203)(109,207)(110,208)
(111,205)(112,206)(113,209)(114,210)(115,212)(116,211)(117,215)(118,216)
(119,213)(120,214)(121,169)(122,170)(123,172)(124,171)(125,175)(126,176)
(127,173)(128,174)(129,177)(130,178)(131,180)(132,179)(133,183)(134,184)
(135,181)(136,182)(137,185)(138,186)(139,188)(140,187)(141,191)(142,192)
(143,189)(144,190)(217,218)(221,224)(222,223)(225,226)(229,232)(230,231)
(233,234)(237,240)(238,239)(241,266)(242,265)(243,267)(244,268)(245,272)
(246,271)(247,270)(248,269)(249,274)(250,273)(251,275)(252,276)(253,280)
(254,279)(255,278)(256,277)(257,282)(258,281)(259,283)(260,284)(261,288)
(262,287)(263,286)(264,285)(289,362)(290,361)(291,363)(292,364)(293,368)
(294,367)(295,366)(296,365)(297,370)(298,369)(299,371)(300,372)(301,376)
(302,375)(303,374)(304,373)(305,378)(306,377)(307,379)(308,380)(309,384)
(310,383)(311,382)(312,381)(313,410)(314,409)(315,411)(316,412)(317,416)
(318,415)(319,414)(320,413)(321,418)(322,417)(323,419)(324,420)(325,424)
(326,423)(327,422)(328,421)(329,426)(330,425)(331,427)(332,428)(333,432)
(334,431)(335,430)(336,429)(337,386)(338,385)(339,387)(340,388)(341,392)
(342,391)(343,390)(344,389)(345,394)(346,393)(347,395)(348,396)(349,400)
(350,399)(351,398)(352,397)(353,402)(354,401)(355,403)(356,404)(357,408)
(358,407)(359,406)(360,405);
s1 := Sym(432)!(  1,393)(  2,394)(  3,398)(  4,397)(  5,396)(  6,395)(  7,400)
(  8,399)(  9,401)( 10,402)( 11,406)( 12,405)( 13,404)( 14,403)( 15,408)
( 16,407)( 17,385)( 18,386)( 19,390)( 20,389)( 21,388)( 22,387)( 23,392)
( 24,391)( 25,377)( 26,378)( 27,382)( 28,381)( 29,380)( 30,379)( 31,384)
( 32,383)( 33,361)( 34,362)( 35,366)( 36,365)( 37,364)( 38,363)( 39,368)
( 40,367)( 41,369)( 42,370)( 43,374)( 44,373)( 45,372)( 46,371)( 47,376)
( 48,375)( 49,409)( 50,410)( 51,414)( 52,413)( 53,412)( 54,411)( 55,416)
( 56,415)( 57,417)( 58,418)( 59,422)( 60,421)( 61,420)( 62,419)( 63,424)
( 64,423)( 65,425)( 66,426)( 67,430)( 68,429)( 69,428)( 70,427)( 71,432)
( 72,431)( 73,321)( 74,322)( 75,326)( 76,325)( 77,324)( 78,323)( 79,328)
( 80,327)( 81,329)( 82,330)( 83,334)( 84,333)( 85,332)( 86,331)( 87,336)
( 88,335)( 89,313)( 90,314)( 91,318)( 92,317)( 93,316)( 94,315)( 95,320)
( 96,319)( 97,305)( 98,306)( 99,310)(100,309)(101,308)(102,307)(103,312)
(104,311)(105,289)(106,290)(107,294)(108,293)(109,292)(110,291)(111,296)
(112,295)(113,297)(114,298)(115,302)(116,301)(117,300)(118,299)(119,304)
(120,303)(121,337)(122,338)(123,342)(124,341)(125,340)(126,339)(127,344)
(128,343)(129,345)(130,346)(131,350)(132,349)(133,348)(134,347)(135,352)
(136,351)(137,353)(138,354)(139,358)(140,357)(141,356)(142,355)(143,360)
(144,359)(145,249)(146,250)(147,254)(148,253)(149,252)(150,251)(151,256)
(152,255)(153,257)(154,258)(155,262)(156,261)(157,260)(158,259)(159,264)
(160,263)(161,241)(162,242)(163,246)(164,245)(165,244)(166,243)(167,248)
(168,247)(169,233)(170,234)(171,238)(172,237)(173,236)(174,235)(175,240)
(176,239)(177,217)(178,218)(179,222)(180,221)(181,220)(182,219)(183,224)
(184,223)(185,225)(186,226)(187,230)(188,229)(189,228)(190,227)(191,232)
(192,231)(193,265)(194,266)(195,270)(196,269)(197,268)(198,267)(199,272)
(200,271)(201,273)(202,274)(203,278)(204,277)(205,276)(206,275)(207,280)
(208,279)(209,281)(210,282)(211,286)(212,285)(213,284)(214,283)(215,288)
(216,287);
s2 := Sym(432)!(  1,219)(  2,220)(  3,217)(  4,218)(  5,222)(  6,221)(  7,223)
(  8,224)(  9,235)( 10,236)( 11,233)( 12,234)( 13,238)( 14,237)( 15,239)
( 16,240)( 17,227)( 18,228)( 19,225)( 20,226)( 21,230)( 22,229)( 23,231)
( 24,232)( 25,243)( 26,244)( 27,241)( 28,242)( 29,246)( 30,245)( 31,247)
( 32,248)( 33,259)( 34,260)( 35,257)( 36,258)( 37,262)( 38,261)( 39,263)
( 40,264)( 41,251)( 42,252)( 43,249)( 44,250)( 45,254)( 46,253)( 47,255)
( 48,256)( 49,267)( 50,268)( 51,265)( 52,266)( 53,270)( 54,269)( 55,271)
( 56,272)( 57,283)( 58,284)( 59,281)( 60,282)( 61,286)( 62,285)( 63,287)
( 64,288)( 65,275)( 66,276)( 67,273)( 68,274)( 69,278)( 70,277)( 71,279)
( 72,280)( 73,363)( 74,364)( 75,361)( 76,362)( 77,366)( 78,365)( 79,367)
( 80,368)( 81,379)( 82,380)( 83,377)( 84,378)( 85,382)( 86,381)( 87,383)
( 88,384)( 89,371)( 90,372)( 91,369)( 92,370)( 93,374)( 94,373)( 95,375)
( 96,376)( 97,387)( 98,388)( 99,385)(100,386)(101,390)(102,389)(103,391)
(104,392)(105,403)(106,404)(107,401)(108,402)(109,406)(110,405)(111,407)
(112,408)(113,395)(114,396)(115,393)(116,394)(117,398)(118,397)(119,399)
(120,400)(121,411)(122,412)(123,409)(124,410)(125,414)(126,413)(127,415)
(128,416)(129,427)(130,428)(131,425)(132,426)(133,430)(134,429)(135,431)
(136,432)(137,419)(138,420)(139,417)(140,418)(141,422)(142,421)(143,423)
(144,424)(145,291)(146,292)(147,289)(148,290)(149,294)(150,293)(151,295)
(152,296)(153,307)(154,308)(155,305)(156,306)(157,310)(158,309)(159,311)
(160,312)(161,299)(162,300)(163,297)(164,298)(165,302)(166,301)(167,303)
(168,304)(169,315)(170,316)(171,313)(172,314)(173,318)(174,317)(175,319)
(176,320)(177,331)(178,332)(179,329)(180,330)(181,334)(182,333)(183,335)
(184,336)(185,323)(186,324)(187,321)(188,322)(189,326)(190,325)(191,327)
(192,328)(193,339)(194,340)(195,337)(196,338)(197,342)(198,341)(199,343)
(200,344)(201,355)(202,356)(203,353)(204,354)(205,358)(206,357)(207,359)
(208,360)(209,347)(210,348)(211,345)(212,346)(213,350)(214,349)(215,351)
(216,352);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope