Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*768f
if this polytope has a name.
Group : SmallGroup(768,1088555)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 64, 192, 64
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
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}*384c
   4-fold quotients : {6,6}*192a
   8-fold quotients : {6,6}*96
   16-fold quotients : {3,6}*48, {6,3}*48
   32-fold quotients : {3,3}*24
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope