Polytope of Type {6,68}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,68}*1632
if this polytope has a name.
Group : SmallGroup(1632,1195)
Rank : 3
Schlafli Type : {6,68}
Number of vertices, edges, etc : 12, 408, 136
Order of s0s1s2 : 102
Order of s0s1s2s1 : 4
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 : {6,68}*816b
4-fold quotients : {6,34}*408
12-fold quotients : {2,34}*136
17-fold quotients : {6,4}*96
24-fold quotients : {2,17}*68
34-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
68-fold quotients : {3,4}*24, {6,2}*24
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 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)
( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)( 63, 64)
( 67, 68)( 69,137)( 70,138)( 71,140)( 72,139)( 73,141)( 74,142)( 75,144)
( 76,143)( 77,145)( 78,146)( 79,148)( 80,147)( 81,149)( 82,150)( 83,152)
( 84,151)( 85,153)( 86,154)( 87,156)( 88,155)( 89,157)( 90,158)( 91,160)
( 92,159)( 93,161)( 94,162)( 95,164)( 96,163)( 97,165)( 98,166)( 99,168)
(100,167)(101,169)(102,170)(103,172)(104,171)(105,173)(106,174)(107,176)
(108,175)(109,177)(110,178)(111,180)(112,179)(113,181)(114,182)(115,184)
(116,183)(117,185)(118,186)(119,188)(120,187)(121,189)(122,190)(123,192)
(124,191)(125,193)(126,194)(127,196)(128,195)(129,197)(130,198)(131,200)
(132,199)(133,201)(134,202)(135,204)(136,203)(207,208)(211,212)(215,216)
(219,220)(223,224)(227,228)(231,232)(235,236)(239,240)(243,244)(247,248)
(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(273,341)(274,342)
(275,344)(276,343)(277,345)(278,346)(279,348)(280,347)(281,349)(282,350)
(283,352)(284,351)(285,353)(286,354)(287,356)(288,355)(289,357)(290,358)
(291,360)(292,359)(293,361)(294,362)(295,364)(296,363)(297,365)(298,366)
(299,368)(300,367)(301,369)(302,370)(303,372)(304,371)(305,373)(306,374)
(307,376)(308,375)(309,377)(310,378)(311,380)(312,379)(313,381)(314,382)
(315,384)(316,383)(317,385)(318,386)(319,388)(320,387)(321,389)(322,390)
(323,392)(324,391)(325,393)(326,394)(327,396)(328,395)(329,397)(330,398)
(331,400)(332,399)(333,401)(334,402)(335,404)(336,403)(337,405)(338,406)
(339,408)(340,407);;
s1 := ( 1, 69)( 2, 72)( 3, 71)( 4, 70)( 5,133)( 6,136)( 7,135)( 8,134)
( 9,129)( 10,132)( 11,131)( 12,130)( 13,125)( 14,128)( 15,127)( 16,126)
( 17,121)( 18,124)( 19,123)( 20,122)( 21,117)( 22,120)( 23,119)( 24,118)
( 25,113)( 26,116)( 27,115)( 28,114)( 29,109)( 30,112)( 31,111)( 32,110)
( 33,105)( 34,108)( 35,107)( 36,106)( 37,101)( 38,104)( 39,103)( 40,102)
( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45, 93)( 46, 96)( 47, 95)( 48, 94)
( 49, 89)( 50, 92)( 51, 91)( 52, 90)( 53, 85)( 54, 88)( 55, 87)( 56, 86)
( 57, 81)( 58, 84)( 59, 83)( 60, 82)( 61, 77)( 62, 80)( 63, 79)( 64, 78)
( 65, 73)( 66, 76)( 67, 75)( 68, 74)(138,140)(141,201)(142,204)(143,203)
(144,202)(145,197)(146,200)(147,199)(148,198)(149,193)(150,196)(151,195)
(152,194)(153,189)(154,192)(155,191)(156,190)(157,185)(158,188)(159,187)
(160,186)(161,181)(162,184)(163,183)(164,182)(165,177)(166,180)(167,179)
(168,178)(169,173)(170,176)(171,175)(172,174)(205,273)(206,276)(207,275)
(208,274)(209,337)(210,340)(211,339)(212,338)(213,333)(214,336)(215,335)
(216,334)(217,329)(218,332)(219,331)(220,330)(221,325)(222,328)(223,327)
(224,326)(225,321)(226,324)(227,323)(228,322)(229,317)(230,320)(231,319)
(232,318)(233,313)(234,316)(235,315)(236,314)(237,309)(238,312)(239,311)
(240,310)(241,305)(242,308)(243,307)(244,306)(245,301)(246,304)(247,303)
(248,302)(249,297)(250,300)(251,299)(252,298)(253,293)(254,296)(255,295)
(256,294)(257,289)(258,292)(259,291)(260,290)(261,285)(262,288)(263,287)
(264,286)(265,281)(266,284)(267,283)(268,282)(269,277)(270,280)(271,279)
(272,278)(342,344)(345,405)(346,408)(347,407)(348,406)(349,401)(350,404)
(351,403)(352,402)(353,397)(354,400)(355,399)(356,398)(357,393)(358,396)
(359,395)(360,394)(361,389)(362,392)(363,391)(364,390)(365,385)(366,388)
(367,387)(368,386)(369,381)(370,384)(371,383)(372,382)(373,377)(374,380)
(375,379)(376,378);;
s2 := ( 1,210)( 2,209)( 3,212)( 4,211)( 5,206)( 6,205)( 7,208)( 8,207)
( 9,270)( 10,269)( 11,272)( 12,271)( 13,266)( 14,265)( 15,268)( 16,267)
( 17,262)( 18,261)( 19,264)( 20,263)( 21,258)( 22,257)( 23,260)( 24,259)
( 25,254)( 26,253)( 27,256)( 28,255)( 29,250)( 30,249)( 31,252)( 32,251)
( 33,246)( 34,245)( 35,248)( 36,247)( 37,242)( 38,241)( 39,244)( 40,243)
( 41,238)( 42,237)( 43,240)( 44,239)( 45,234)( 46,233)( 47,236)( 48,235)
( 49,230)( 50,229)( 51,232)( 52,231)( 53,226)( 54,225)( 55,228)( 56,227)
( 57,222)( 58,221)( 59,224)( 60,223)( 61,218)( 62,217)( 63,220)( 64,219)
( 65,214)( 66,213)( 67,216)( 68,215)( 69,278)( 70,277)( 71,280)( 72,279)
( 73,274)( 74,273)( 75,276)( 76,275)( 77,338)( 78,337)( 79,340)( 80,339)
( 81,334)( 82,333)( 83,336)( 84,335)( 85,330)( 86,329)( 87,332)( 88,331)
( 89,326)( 90,325)( 91,328)( 92,327)( 93,322)( 94,321)( 95,324)( 96,323)
( 97,318)( 98,317)( 99,320)(100,319)(101,314)(102,313)(103,316)(104,315)
(105,310)(106,309)(107,312)(108,311)(109,306)(110,305)(111,308)(112,307)
(113,302)(114,301)(115,304)(116,303)(117,298)(118,297)(119,300)(120,299)
(121,294)(122,293)(123,296)(124,295)(125,290)(126,289)(127,292)(128,291)
(129,286)(130,285)(131,288)(132,287)(133,282)(134,281)(135,284)(136,283)
(137,346)(138,345)(139,348)(140,347)(141,342)(142,341)(143,344)(144,343)
(145,406)(146,405)(147,408)(148,407)(149,402)(150,401)(151,404)(152,403)
(153,398)(154,397)(155,400)(156,399)(157,394)(158,393)(159,396)(160,395)
(161,390)(162,389)(163,392)(164,391)(165,386)(166,385)(167,388)(168,387)
(169,382)(170,381)(171,384)(172,383)(173,378)(174,377)(175,380)(176,379)
(177,374)(178,373)(179,376)(180,375)(181,370)(182,369)(183,372)(184,371)
(185,366)(186,365)(187,368)(188,367)(189,362)(190,361)(191,364)(192,363)
(193,358)(194,357)(195,360)(196,359)(197,354)(198,353)(199,356)(200,355)
(201,350)(202,349)(203,352)(204,351);;
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,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(408)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)
( 31, 32)( 35, 36)( 39, 40)( 43, 44)( 47, 48)( 51, 52)( 55, 56)( 59, 60)
( 63, 64)( 67, 68)( 69,137)( 70,138)( 71,140)( 72,139)( 73,141)( 74,142)
( 75,144)( 76,143)( 77,145)( 78,146)( 79,148)( 80,147)( 81,149)( 82,150)
( 83,152)( 84,151)( 85,153)( 86,154)( 87,156)( 88,155)( 89,157)( 90,158)
( 91,160)( 92,159)( 93,161)( 94,162)( 95,164)( 96,163)( 97,165)( 98,166)
( 99,168)(100,167)(101,169)(102,170)(103,172)(104,171)(105,173)(106,174)
(107,176)(108,175)(109,177)(110,178)(111,180)(112,179)(113,181)(114,182)
(115,184)(116,183)(117,185)(118,186)(119,188)(120,187)(121,189)(122,190)
(123,192)(124,191)(125,193)(126,194)(127,196)(128,195)(129,197)(130,198)
(131,200)(132,199)(133,201)(134,202)(135,204)(136,203)(207,208)(211,212)
(215,216)(219,220)(223,224)(227,228)(231,232)(235,236)(239,240)(243,244)
(247,248)(251,252)(255,256)(259,260)(263,264)(267,268)(271,272)(273,341)
(274,342)(275,344)(276,343)(277,345)(278,346)(279,348)(280,347)(281,349)
(282,350)(283,352)(284,351)(285,353)(286,354)(287,356)(288,355)(289,357)
(290,358)(291,360)(292,359)(293,361)(294,362)(295,364)(296,363)(297,365)
(298,366)(299,368)(300,367)(301,369)(302,370)(303,372)(304,371)(305,373)
(306,374)(307,376)(308,375)(309,377)(310,378)(311,380)(312,379)(313,381)
(314,382)(315,384)(316,383)(317,385)(318,386)(319,388)(320,387)(321,389)
(322,390)(323,392)(324,391)(325,393)(326,394)(327,396)(328,395)(329,397)
(330,398)(331,400)(332,399)(333,401)(334,402)(335,404)(336,403)(337,405)
(338,406)(339,408)(340,407);
s1 := Sym(408)!( 1, 69)( 2, 72)( 3, 71)( 4, 70)( 5,133)( 6,136)( 7,135)
( 8,134)( 9,129)( 10,132)( 11,131)( 12,130)( 13,125)( 14,128)( 15,127)
( 16,126)( 17,121)( 18,124)( 19,123)( 20,122)( 21,117)( 22,120)( 23,119)
( 24,118)( 25,113)( 26,116)( 27,115)( 28,114)( 29,109)( 30,112)( 31,111)
( 32,110)( 33,105)( 34,108)( 35,107)( 36,106)( 37,101)( 38,104)( 39,103)
( 40,102)( 41, 97)( 42,100)( 43, 99)( 44, 98)( 45, 93)( 46, 96)( 47, 95)
( 48, 94)( 49, 89)( 50, 92)( 51, 91)( 52, 90)( 53, 85)( 54, 88)( 55, 87)
( 56, 86)( 57, 81)( 58, 84)( 59, 83)( 60, 82)( 61, 77)( 62, 80)( 63, 79)
( 64, 78)( 65, 73)( 66, 76)( 67, 75)( 68, 74)(138,140)(141,201)(142,204)
(143,203)(144,202)(145,197)(146,200)(147,199)(148,198)(149,193)(150,196)
(151,195)(152,194)(153,189)(154,192)(155,191)(156,190)(157,185)(158,188)
(159,187)(160,186)(161,181)(162,184)(163,183)(164,182)(165,177)(166,180)
(167,179)(168,178)(169,173)(170,176)(171,175)(172,174)(205,273)(206,276)
(207,275)(208,274)(209,337)(210,340)(211,339)(212,338)(213,333)(214,336)
(215,335)(216,334)(217,329)(218,332)(219,331)(220,330)(221,325)(222,328)
(223,327)(224,326)(225,321)(226,324)(227,323)(228,322)(229,317)(230,320)
(231,319)(232,318)(233,313)(234,316)(235,315)(236,314)(237,309)(238,312)
(239,311)(240,310)(241,305)(242,308)(243,307)(244,306)(245,301)(246,304)
(247,303)(248,302)(249,297)(250,300)(251,299)(252,298)(253,293)(254,296)
(255,295)(256,294)(257,289)(258,292)(259,291)(260,290)(261,285)(262,288)
(263,287)(264,286)(265,281)(266,284)(267,283)(268,282)(269,277)(270,280)
(271,279)(272,278)(342,344)(345,405)(346,408)(347,407)(348,406)(349,401)
(350,404)(351,403)(352,402)(353,397)(354,400)(355,399)(356,398)(357,393)
(358,396)(359,395)(360,394)(361,389)(362,392)(363,391)(364,390)(365,385)
(366,388)(367,387)(368,386)(369,381)(370,384)(371,383)(372,382)(373,377)
(374,380)(375,379)(376,378);
s2 := Sym(408)!( 1,210)( 2,209)( 3,212)( 4,211)( 5,206)( 6,205)( 7,208)
( 8,207)( 9,270)( 10,269)( 11,272)( 12,271)( 13,266)( 14,265)( 15,268)
( 16,267)( 17,262)( 18,261)( 19,264)( 20,263)( 21,258)( 22,257)( 23,260)
( 24,259)( 25,254)( 26,253)( 27,256)( 28,255)( 29,250)( 30,249)( 31,252)
( 32,251)( 33,246)( 34,245)( 35,248)( 36,247)( 37,242)( 38,241)( 39,244)
( 40,243)( 41,238)( 42,237)( 43,240)( 44,239)( 45,234)( 46,233)( 47,236)
( 48,235)( 49,230)( 50,229)( 51,232)( 52,231)( 53,226)( 54,225)( 55,228)
( 56,227)( 57,222)( 58,221)( 59,224)( 60,223)( 61,218)( 62,217)( 63,220)
( 64,219)( 65,214)( 66,213)( 67,216)( 68,215)( 69,278)( 70,277)( 71,280)
( 72,279)( 73,274)( 74,273)( 75,276)( 76,275)( 77,338)( 78,337)( 79,340)
( 80,339)( 81,334)( 82,333)( 83,336)( 84,335)( 85,330)( 86,329)( 87,332)
( 88,331)( 89,326)( 90,325)( 91,328)( 92,327)( 93,322)( 94,321)( 95,324)
( 96,323)( 97,318)( 98,317)( 99,320)(100,319)(101,314)(102,313)(103,316)
(104,315)(105,310)(106,309)(107,312)(108,311)(109,306)(110,305)(111,308)
(112,307)(113,302)(114,301)(115,304)(116,303)(117,298)(118,297)(119,300)
(120,299)(121,294)(122,293)(123,296)(124,295)(125,290)(126,289)(127,292)
(128,291)(129,286)(130,285)(131,288)(132,287)(133,282)(134,281)(135,284)
(136,283)(137,346)(138,345)(139,348)(140,347)(141,342)(142,341)(143,344)
(144,343)(145,406)(146,405)(147,408)(148,407)(149,402)(150,401)(151,404)
(152,403)(153,398)(154,397)(155,400)(156,399)(157,394)(158,393)(159,396)
(160,395)(161,390)(162,389)(163,392)(164,391)(165,386)(166,385)(167,388)
(168,387)(169,382)(170,381)(171,384)(172,383)(173,378)(174,377)(175,380)
(176,379)(177,374)(178,373)(179,376)(180,375)(181,370)(182,369)(183,372)
(184,371)(185,366)(186,365)(187,368)(188,367)(189,362)(190,361)(191,364)
(192,363)(193,358)(194,357)(195,360)(196,359)(197,354)(198,353)(199,356)
(200,355)(201,350)(202,349)(203,352)(204,351);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*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*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1 >;
References : None.
to this polytope