Polytope of Type {12,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*1728k
if this polytope has a name.
Group : SmallGroup(1728,30242)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 72, 432, 72
Order of s0s1s2 : 12
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 : {12,6}*864a
3-fold quotients : {12,12}*576g
4-fold quotients : {6,12}*432a, {12,3}*432
6-fold quotients : {12,6}*288b
8-fold quotients : {6,6}*216a
9-fold quotients : {4,12}*192b
12-fold quotients : {6,12}*144b, {12,3}*144
16-fold quotients : {6,3}*108
18-fold quotients : {4,12}*96b, {4,12}*96c, {4,6}*96
24-fold quotients : {6,6}*72b
36-fold quotients : {2,12}*48, {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {6,3}*36
72-fold quotients : {4,3}*24, {2,6}*24
108-fold quotients : {2,4}*16
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 := ( 1, 3)( 2, 4)( 5, 11)( 6, 12)( 7, 9)( 8, 10)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)( 22, 32)
( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)( 44, 46)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)( 56, 70)
( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)( 78, 84)
( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)( 90,108)
( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)(124,134)
(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)(132,138)
(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)(158,172)
(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)(166,176)
(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)(188,190)
(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)(200,214)
(201,211)(202,212)(203,209)(204,210)(217,219)(218,220)(221,227)(222,228)
(223,225)(224,226)(229,243)(230,244)(231,241)(232,242)(233,251)(234,252)
(235,249)(236,250)(237,247)(238,248)(239,245)(240,246)(253,255)(254,256)
(257,263)(258,264)(259,261)(260,262)(265,279)(266,280)(267,277)(268,278)
(269,287)(270,288)(271,285)(272,286)(273,283)(274,284)(275,281)(276,282)
(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,315)(302,316)
(303,313)(304,314)(305,323)(306,324)(307,321)(308,322)(309,319)(310,320)
(311,317)(312,318)(325,327)(326,328)(329,335)(330,336)(331,333)(332,334)
(337,351)(338,352)(339,349)(340,350)(341,359)(342,360)(343,357)(344,358)
(345,355)(346,356)(347,353)(348,354)(361,363)(362,364)(365,371)(366,372)
(367,369)(368,370)(373,387)(374,388)(375,385)(376,386)(377,395)(378,396)
(379,393)(380,394)(381,391)(382,392)(383,389)(384,390)(397,399)(398,400)
(401,407)(402,408)(403,405)(404,406)(409,423)(410,424)(411,421)(412,422)
(413,431)(414,432)(415,429)(416,430)(417,427)(418,428)(419,425)(420,426);;
s1 := ( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 17)( 6, 18)( 7, 20)( 8, 19)
( 9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 85)
( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 93)
( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 77)
( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)
( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)(113,125)
(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)(135,136)
(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)(150,198)
(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)(158,182)
(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)(166,190)
(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)
(175,212)(176,211)(177,213)(178,214)(179,216)(180,215)(217,337)(218,338)
(219,340)(220,339)(221,341)(222,342)(223,344)(224,343)(225,345)(226,346)
(227,348)(228,347)(229,325)(230,326)(231,328)(232,327)(233,329)(234,330)
(235,332)(236,331)(237,333)(238,334)(239,336)(240,335)(241,349)(242,350)
(243,352)(244,351)(245,353)(246,354)(247,356)(248,355)(249,357)(250,358)
(251,360)(252,359)(253,409)(254,410)(255,412)(256,411)(257,413)(258,414)
(259,416)(260,415)(261,417)(262,418)(263,420)(264,419)(265,397)(266,398)
(267,400)(268,399)(269,401)(270,402)(271,404)(272,403)(273,405)(274,406)
(275,408)(276,407)(277,421)(278,422)(279,424)(280,423)(281,425)(282,426)
(283,428)(284,427)(285,429)(286,430)(287,432)(288,431)(289,373)(290,374)
(291,376)(292,375)(293,377)(294,378)(295,380)(296,379)(297,381)(298,382)
(299,384)(300,383)(301,361)(302,362)(303,364)(304,363)(305,365)(306,366)
(307,368)(308,367)(309,369)(310,370)(311,372)(312,371)(313,385)(314,386)
(315,388)(316,387)(317,389)(318,390)(319,392)(320,391)(321,393)(322,394)
(323,396)(324,395);;
s2 := ( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)( 8,258)
( 9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)( 16,286)
( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)( 24,282)
( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)( 40,218)
( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)( 48,226)
( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)( 56,242)
( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)( 64,234)
( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)( 72,230)
( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)( 80,294)
( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)( 88,322)
( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)( 96,318)
( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)(104,310)
(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)(112,362)
(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)(120,370)
(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)(128,386)
(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)(136,378)
(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)(144,374)
(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)(152,330)
(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)(160,358)
(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)(168,354)
(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)(176,346)
(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)(184,398)
(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)(192,406)
(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)(200,422)
(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)(208,414)
(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)(216,410);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 1, 3)( 2, 4)( 5, 11)( 6, 12)( 7, 9)( 8, 10)( 13, 27)
( 14, 28)( 15, 25)( 16, 26)( 17, 35)( 18, 36)( 19, 33)( 20, 34)( 21, 31)
( 22, 32)( 23, 29)( 24, 30)( 37, 39)( 38, 40)( 41, 47)( 42, 48)( 43, 45)
( 44, 46)( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 71)( 54, 72)( 55, 69)
( 56, 70)( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 73, 75)( 74, 76)( 77, 83)
( 78, 84)( 79, 81)( 80, 82)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,107)
( 90,108)( 91,105)( 92,106)( 93,103)( 94,104)( 95,101)( 96,102)(109,111)
(110,112)(113,119)(114,120)(115,117)(116,118)(121,135)(122,136)(123,133)
(124,134)(125,143)(126,144)(127,141)(128,142)(129,139)(130,140)(131,137)
(132,138)(145,147)(146,148)(149,155)(150,156)(151,153)(152,154)(157,171)
(158,172)(159,169)(160,170)(161,179)(162,180)(163,177)(164,178)(165,175)
(166,176)(167,173)(168,174)(181,183)(182,184)(185,191)(186,192)(187,189)
(188,190)(193,207)(194,208)(195,205)(196,206)(197,215)(198,216)(199,213)
(200,214)(201,211)(202,212)(203,209)(204,210)(217,219)(218,220)(221,227)
(222,228)(223,225)(224,226)(229,243)(230,244)(231,241)(232,242)(233,251)
(234,252)(235,249)(236,250)(237,247)(238,248)(239,245)(240,246)(253,255)
(254,256)(257,263)(258,264)(259,261)(260,262)(265,279)(266,280)(267,277)
(268,278)(269,287)(270,288)(271,285)(272,286)(273,283)(274,284)(275,281)
(276,282)(289,291)(290,292)(293,299)(294,300)(295,297)(296,298)(301,315)
(302,316)(303,313)(304,314)(305,323)(306,324)(307,321)(308,322)(309,319)
(310,320)(311,317)(312,318)(325,327)(326,328)(329,335)(330,336)(331,333)
(332,334)(337,351)(338,352)(339,349)(340,350)(341,359)(342,360)(343,357)
(344,358)(345,355)(346,356)(347,353)(348,354)(361,363)(362,364)(365,371)
(366,372)(367,369)(368,370)(373,387)(374,388)(375,385)(376,386)(377,395)
(378,396)(379,393)(380,394)(381,391)(382,392)(383,389)(384,390)(397,399)
(398,400)(401,407)(402,408)(403,405)(404,406)(409,423)(410,424)(411,421)
(412,422)(413,431)(414,432)(415,429)(416,430)(417,427)(418,428)(419,425)
(420,426);
s1 := Sym(432)!( 1, 13)( 2, 14)( 3, 16)( 4, 15)( 5, 17)( 6, 18)( 7, 20)
( 8, 19)( 9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)
( 37, 85)( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)
( 45, 93)( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)
( 53, 77)( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)
( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)
( 69,105)( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)
(113,125)(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)
(135,136)(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)
(150,198)(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)
(158,182)(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)
(166,190)(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)
(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215)(217,337)
(218,338)(219,340)(220,339)(221,341)(222,342)(223,344)(224,343)(225,345)
(226,346)(227,348)(228,347)(229,325)(230,326)(231,328)(232,327)(233,329)
(234,330)(235,332)(236,331)(237,333)(238,334)(239,336)(240,335)(241,349)
(242,350)(243,352)(244,351)(245,353)(246,354)(247,356)(248,355)(249,357)
(250,358)(251,360)(252,359)(253,409)(254,410)(255,412)(256,411)(257,413)
(258,414)(259,416)(260,415)(261,417)(262,418)(263,420)(264,419)(265,397)
(266,398)(267,400)(268,399)(269,401)(270,402)(271,404)(272,403)(273,405)
(274,406)(275,408)(276,407)(277,421)(278,422)(279,424)(280,423)(281,425)
(282,426)(283,428)(284,427)(285,429)(286,430)(287,432)(288,431)(289,373)
(290,374)(291,376)(292,375)(293,377)(294,378)(295,380)(296,379)(297,381)
(298,382)(299,384)(300,383)(301,361)(302,362)(303,364)(304,363)(305,365)
(306,366)(307,368)(308,367)(309,369)(310,370)(311,372)(312,371)(313,385)
(314,386)(315,388)(316,387)(317,389)(318,390)(319,392)(320,391)(321,393)
(322,394)(323,396)(324,395);
s2 := Sym(432)!( 1,253)( 2,256)( 3,255)( 4,254)( 5,257)( 6,260)( 7,259)
( 8,258)( 9,261)( 10,264)( 11,263)( 12,262)( 13,285)( 14,288)( 15,287)
( 16,286)( 17,277)( 18,280)( 19,279)( 20,278)( 21,281)( 22,284)( 23,283)
( 24,282)( 25,269)( 26,272)( 27,271)( 28,270)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,265)( 34,268)( 35,267)( 36,266)( 37,217)( 38,220)( 39,219)
( 40,218)( 41,221)( 42,224)( 43,223)( 44,222)( 45,225)( 46,228)( 47,227)
( 48,226)( 49,249)( 50,252)( 51,251)( 52,250)( 53,241)( 54,244)( 55,243)
( 56,242)( 57,245)( 58,248)( 59,247)( 60,246)( 61,233)( 62,236)( 63,235)
( 64,234)( 65,237)( 66,240)( 67,239)( 68,238)( 69,229)( 70,232)( 71,231)
( 72,230)( 73,289)( 74,292)( 75,291)( 76,290)( 77,293)( 78,296)( 79,295)
( 80,294)( 81,297)( 82,300)( 83,299)( 84,298)( 85,321)( 86,324)( 87,323)
( 88,322)( 89,313)( 90,316)( 91,315)( 92,314)( 93,317)( 94,320)( 95,319)
( 96,318)( 97,305)( 98,308)( 99,307)(100,306)(101,309)(102,312)(103,311)
(104,310)(105,301)(106,304)(107,303)(108,302)(109,361)(110,364)(111,363)
(112,362)(113,365)(114,368)(115,367)(116,366)(117,369)(118,372)(119,371)
(120,370)(121,393)(122,396)(123,395)(124,394)(125,385)(126,388)(127,387)
(128,386)(129,389)(130,392)(131,391)(132,390)(133,377)(134,380)(135,379)
(136,378)(137,381)(138,384)(139,383)(140,382)(141,373)(142,376)(143,375)
(144,374)(145,325)(146,328)(147,327)(148,326)(149,329)(150,332)(151,331)
(152,330)(153,333)(154,336)(155,335)(156,334)(157,357)(158,360)(159,359)
(160,358)(161,349)(162,352)(163,351)(164,350)(165,353)(166,356)(167,355)
(168,354)(169,341)(170,344)(171,343)(172,342)(173,345)(174,348)(175,347)
(176,346)(177,337)(178,340)(179,339)(180,338)(181,397)(182,400)(183,399)
(184,398)(185,401)(186,404)(187,403)(188,402)(189,405)(190,408)(191,407)
(192,406)(193,429)(194,432)(195,431)(196,430)(197,421)(198,424)(199,423)
(200,422)(201,425)(202,428)(203,427)(204,426)(205,413)(206,416)(207,415)
(208,414)(209,417)(210,420)(211,419)(212,418)(213,409)(214,412)(215,411)
(216,410);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope