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