Polytope of Type {12,68}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,68}*1632
Also Known As : {12,68|2}. if this polytope has another name.
Group : SmallGroup(1632,548)
Rank : 3
Schlafli Type : {12,68}
Number of vertices, edges, etc : 12, 408, 68
Order of s0s1s2 : 204
Order of s0s1s2s1 : 2
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 : {12,34}*816, {6,68}*816a
3-fold quotients : {4,68}*544
4-fold quotients : {6,34}*408
6-fold quotients : {2,68}*272, {4,34}*272
12-fold quotients : {2,34}*136
17-fold quotients : {12,4}*96a
24-fold quotients : {2,17}*68
34-fold quotients : {12,2}*48, {6,4}*48a
51-fold quotients : {4,4}*32
68-fold quotients : {6,2}*24
102-fold quotients : {2,4}*16, {4,2}*16
136-fold quotients : {3,2}*12
204-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)( 25, 42)
( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)( 33, 50)
( 34, 51)( 69, 86)( 70, 87)( 71, 88)( 72, 89)( 73, 90)( 74, 91)( 75, 92)
( 76, 93)( 77, 94)( 78, 95)( 79, 96)( 80, 97)( 81, 98)( 82, 99)( 83,100)
( 84,101)( 85,102)(120,137)(121,138)(122,139)(123,140)(124,141)(125,142)
(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)(133,150)
(134,151)(135,152)(136,153)(171,188)(172,189)(173,190)(174,191)(175,192)
(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)(183,200)
(184,201)(185,202)(186,203)(187,204)(205,307)(206,308)(207,309)(208,310)
(209,311)(210,312)(211,313)(212,314)(213,315)(214,316)(215,317)(216,318)
(217,319)(218,320)(219,321)(220,322)(221,323)(222,341)(223,342)(224,343)
(225,344)(226,345)(227,346)(228,347)(229,348)(230,349)(231,350)(232,351)
(233,352)(234,353)(235,354)(236,355)(237,356)(238,357)(239,324)(240,325)
(241,326)(242,327)(243,328)(244,329)(245,330)(246,331)(247,332)(248,333)
(249,334)(250,335)(251,336)(252,337)(253,338)(254,339)(255,340)(256,358)
(257,359)(258,360)(259,361)(260,362)(261,363)(262,364)(263,365)(264,366)
(265,367)(266,368)(267,369)(268,370)(269,371)(270,372)(271,373)(272,374)
(273,392)(274,393)(275,394)(276,395)(277,396)(278,397)(279,398)(280,399)
(281,400)(282,401)(283,402)(284,403)(285,404)(286,405)(287,406)(288,407)
(289,408)(290,375)(291,376)(292,377)(293,378)(294,379)(295,380)(296,381)
(297,382)(298,383)(299,384)(300,385)(301,386)(302,387)(303,388)(304,389)
(305,390)(306,391);;
s1 := ( 1,222)( 2,238)( 3,237)( 4,236)( 5,235)( 6,234)( 7,233)( 8,232)
( 9,231)( 10,230)( 11,229)( 12,228)( 13,227)( 14,226)( 15,225)( 16,224)
( 17,223)( 18,205)( 19,221)( 20,220)( 21,219)( 22,218)( 23,217)( 24,216)
( 25,215)( 26,214)( 27,213)( 28,212)( 29,211)( 30,210)( 31,209)( 32,208)
( 33,207)( 34,206)( 35,239)( 36,255)( 37,254)( 38,253)( 39,252)( 40,251)
( 41,250)( 42,249)( 43,248)( 44,247)( 45,246)( 46,245)( 47,244)( 48,243)
( 49,242)( 50,241)( 51,240)( 52,273)( 53,289)( 54,288)( 55,287)( 56,286)
( 57,285)( 58,284)( 59,283)( 60,282)( 61,281)( 62,280)( 63,279)( 64,278)
( 65,277)( 66,276)( 67,275)( 68,274)( 69,256)( 70,272)( 71,271)( 72,270)
( 73,269)( 74,268)( 75,267)( 76,266)( 77,265)( 78,264)( 79,263)( 80,262)
( 81,261)( 82,260)( 83,259)( 84,258)( 85,257)( 86,290)( 87,306)( 88,305)
( 89,304)( 90,303)( 91,302)( 92,301)( 93,300)( 94,299)( 95,298)( 96,297)
( 97,296)( 98,295)( 99,294)(100,293)(101,292)(102,291)(103,324)(104,340)
(105,339)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)(112,332)
(113,331)(114,330)(115,329)(116,328)(117,327)(118,326)(119,325)(120,307)
(121,323)(122,322)(123,321)(124,320)(125,319)(126,318)(127,317)(128,316)
(129,315)(130,314)(131,313)(132,312)(133,311)(134,310)(135,309)(136,308)
(137,341)(138,357)(139,356)(140,355)(141,354)(142,353)(143,352)(144,351)
(145,350)(146,349)(147,348)(148,347)(149,346)(150,345)(151,344)(152,343)
(153,342)(154,375)(155,391)(156,390)(157,389)(158,388)(159,387)(160,386)
(161,385)(162,384)(163,383)(164,382)(165,381)(166,380)(167,379)(168,378)
(169,377)(170,376)(171,358)(172,374)(173,373)(174,372)(175,371)(176,370)
(177,369)(178,368)(179,367)(180,366)(181,365)(182,364)(183,363)(184,362)
(185,361)(186,360)(187,359)(188,392)(189,408)(190,407)(191,406)(192,405)
(193,404)(194,403)(195,402)(196,401)(197,400)(198,399)(199,398)(200,397)
(201,396)(202,395)(203,394)(204,393);;
s2 := ( 1, 2)( 3, 17)( 4, 16)( 5, 15)( 6, 14)( 7, 13)( 8, 12)( 9, 11)
( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)
( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)( 43, 45)
( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)( 60, 62)
( 69, 70)( 71, 85)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)
( 86, 87)( 88,102)( 89,101)( 90,100)( 91, 99)( 92, 98)( 93, 97)( 94, 96)
(103,104)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(111,113)
(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)
(137,138)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)(145,147)
(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)(162,164)
(171,172)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)(179,181)
(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)(196,198)
(205,257)(206,256)(207,272)(208,271)(209,270)(210,269)(211,268)(212,267)
(213,266)(214,265)(215,264)(216,263)(217,262)(218,261)(219,260)(220,259)
(221,258)(222,274)(223,273)(224,289)(225,288)(226,287)(227,286)(228,285)
(229,284)(230,283)(231,282)(232,281)(233,280)(234,279)(235,278)(236,277)
(237,276)(238,275)(239,291)(240,290)(241,306)(242,305)(243,304)(244,303)
(245,302)(246,301)(247,300)(248,299)(249,298)(250,297)(251,296)(252,295)
(253,294)(254,293)(255,292)(307,359)(308,358)(309,374)(310,373)(311,372)
(312,371)(313,370)(314,369)(315,368)(316,367)(317,366)(318,365)(319,364)
(320,363)(321,362)(322,361)(323,360)(324,376)(325,375)(326,391)(327,390)
(328,389)(329,388)(330,387)(331,386)(332,385)(333,384)(334,383)(335,382)
(336,381)(337,380)(338,379)(339,378)(340,377)(341,393)(342,392)(343,408)
(344,407)(345,406)(346,405)(347,404)(348,403)(349,402)(350,401)(351,400)
(352,399)(353,398)(354,397)(355,396)(356,395)(357,394);;
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*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,
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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(408)!( 18, 35)( 19, 36)( 20, 37)( 21, 38)( 22, 39)( 23, 40)( 24, 41)
( 25, 42)( 26, 43)( 27, 44)( 28, 45)( 29, 46)( 30, 47)( 31, 48)( 32, 49)
( 33, 50)( 34, 51)( 69, 86)( 70, 87)( 71, 88)( 72, 89)( 73, 90)( 74, 91)
( 75, 92)( 76, 93)( 77, 94)( 78, 95)( 79, 96)( 80, 97)( 81, 98)( 82, 99)
( 83,100)( 84,101)( 85,102)(120,137)(121,138)(122,139)(123,140)(124,141)
(125,142)(126,143)(127,144)(128,145)(129,146)(130,147)(131,148)(132,149)
(133,150)(134,151)(135,152)(136,153)(171,188)(172,189)(173,190)(174,191)
(175,192)(176,193)(177,194)(178,195)(179,196)(180,197)(181,198)(182,199)
(183,200)(184,201)(185,202)(186,203)(187,204)(205,307)(206,308)(207,309)
(208,310)(209,311)(210,312)(211,313)(212,314)(213,315)(214,316)(215,317)
(216,318)(217,319)(218,320)(219,321)(220,322)(221,323)(222,341)(223,342)
(224,343)(225,344)(226,345)(227,346)(228,347)(229,348)(230,349)(231,350)
(232,351)(233,352)(234,353)(235,354)(236,355)(237,356)(238,357)(239,324)
(240,325)(241,326)(242,327)(243,328)(244,329)(245,330)(246,331)(247,332)
(248,333)(249,334)(250,335)(251,336)(252,337)(253,338)(254,339)(255,340)
(256,358)(257,359)(258,360)(259,361)(260,362)(261,363)(262,364)(263,365)
(264,366)(265,367)(266,368)(267,369)(268,370)(269,371)(270,372)(271,373)
(272,374)(273,392)(274,393)(275,394)(276,395)(277,396)(278,397)(279,398)
(280,399)(281,400)(282,401)(283,402)(284,403)(285,404)(286,405)(287,406)
(288,407)(289,408)(290,375)(291,376)(292,377)(293,378)(294,379)(295,380)
(296,381)(297,382)(298,383)(299,384)(300,385)(301,386)(302,387)(303,388)
(304,389)(305,390)(306,391);
s1 := Sym(408)!( 1,222)( 2,238)( 3,237)( 4,236)( 5,235)( 6,234)( 7,233)
( 8,232)( 9,231)( 10,230)( 11,229)( 12,228)( 13,227)( 14,226)( 15,225)
( 16,224)( 17,223)( 18,205)( 19,221)( 20,220)( 21,219)( 22,218)( 23,217)
( 24,216)( 25,215)( 26,214)( 27,213)( 28,212)( 29,211)( 30,210)( 31,209)
( 32,208)( 33,207)( 34,206)( 35,239)( 36,255)( 37,254)( 38,253)( 39,252)
( 40,251)( 41,250)( 42,249)( 43,248)( 44,247)( 45,246)( 46,245)( 47,244)
( 48,243)( 49,242)( 50,241)( 51,240)( 52,273)( 53,289)( 54,288)( 55,287)
( 56,286)( 57,285)( 58,284)( 59,283)( 60,282)( 61,281)( 62,280)( 63,279)
( 64,278)( 65,277)( 66,276)( 67,275)( 68,274)( 69,256)( 70,272)( 71,271)
( 72,270)( 73,269)( 74,268)( 75,267)( 76,266)( 77,265)( 78,264)( 79,263)
( 80,262)( 81,261)( 82,260)( 83,259)( 84,258)( 85,257)( 86,290)( 87,306)
( 88,305)( 89,304)( 90,303)( 91,302)( 92,301)( 93,300)( 94,299)( 95,298)
( 96,297)( 97,296)( 98,295)( 99,294)(100,293)(101,292)(102,291)(103,324)
(104,340)(105,339)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)
(112,332)(113,331)(114,330)(115,329)(116,328)(117,327)(118,326)(119,325)
(120,307)(121,323)(122,322)(123,321)(124,320)(125,319)(126,318)(127,317)
(128,316)(129,315)(130,314)(131,313)(132,312)(133,311)(134,310)(135,309)
(136,308)(137,341)(138,357)(139,356)(140,355)(141,354)(142,353)(143,352)
(144,351)(145,350)(146,349)(147,348)(148,347)(149,346)(150,345)(151,344)
(152,343)(153,342)(154,375)(155,391)(156,390)(157,389)(158,388)(159,387)
(160,386)(161,385)(162,384)(163,383)(164,382)(165,381)(166,380)(167,379)
(168,378)(169,377)(170,376)(171,358)(172,374)(173,373)(174,372)(175,371)
(176,370)(177,369)(178,368)(179,367)(180,366)(181,365)(182,364)(183,363)
(184,362)(185,361)(186,360)(187,359)(188,392)(189,408)(190,407)(191,406)
(192,405)(193,404)(194,403)(195,402)(196,401)(197,400)(198,399)(199,398)
(200,397)(201,396)(202,395)(203,394)(204,393);
s2 := Sym(408)!( 1, 2)( 3, 17)( 4, 16)( 5, 15)( 6, 14)( 7, 13)( 8, 12)
( 9, 11)( 18, 19)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)
( 26, 28)( 35, 36)( 37, 51)( 38, 50)( 39, 49)( 40, 48)( 41, 47)( 42, 46)
( 43, 45)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)( 59, 63)
( 60, 62)( 69, 70)( 71, 85)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 86, 87)( 88,102)( 89,101)( 90,100)( 91, 99)( 92, 98)( 93, 97)
( 94, 96)(103,104)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)
(111,113)(120,121)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)
(128,130)(137,138)(139,153)(140,152)(141,151)(142,150)(143,149)(144,148)
(145,147)(154,155)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)
(162,164)(171,172)(173,187)(174,186)(175,185)(176,184)(177,183)(178,182)
(179,181)(188,189)(190,204)(191,203)(192,202)(193,201)(194,200)(195,199)
(196,198)(205,257)(206,256)(207,272)(208,271)(209,270)(210,269)(211,268)
(212,267)(213,266)(214,265)(215,264)(216,263)(217,262)(218,261)(219,260)
(220,259)(221,258)(222,274)(223,273)(224,289)(225,288)(226,287)(227,286)
(228,285)(229,284)(230,283)(231,282)(232,281)(233,280)(234,279)(235,278)
(236,277)(237,276)(238,275)(239,291)(240,290)(241,306)(242,305)(243,304)
(244,303)(245,302)(246,301)(247,300)(248,299)(249,298)(250,297)(251,296)
(252,295)(253,294)(254,293)(255,292)(307,359)(308,358)(309,374)(310,373)
(311,372)(312,371)(313,370)(314,369)(315,368)(316,367)(317,366)(318,365)
(319,364)(320,363)(321,362)(322,361)(323,360)(324,376)(325,375)(326,391)
(327,390)(328,389)(329,388)(330,387)(331,386)(332,385)(333,384)(334,383)
(335,382)(336,381)(337,380)(338,379)(339,378)(340,377)(341,393)(342,392)
(343,408)(344,407)(345,406)(346,405)(347,404)(348,403)(349,402)(350,401)
(351,400)(352,399)(353,398)(354,397)(355,396)(356,395)(357,394);
poly := sub<Sym(408)|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*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,
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*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 >;
References : None.
to this polytope