Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*768e
if this polytope has a name.
Group : SmallGroup(768,1088539)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 64, 192, 64
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 6
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}*384a, {6,6}*384b, {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 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)
( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)
( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)
( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)
( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)
( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)
(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)
(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)
(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)
(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)
(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)
(197,199)(198,200)(205,207)(206,208)(209,217)(210,218)(211,219)(212,220)
(213,223)(214,224)(215,221)(216,222)(225,257)(226,258)(227,259)(228,260)
(229,263)(230,264)(231,261)(232,262)(233,265)(234,266)(235,267)(236,268)
(237,271)(238,272)(239,269)(240,270)(241,281)(242,282)(243,283)(244,284)
(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)(251,275)(252,276)
(253,279)(254,280)(255,277)(256,278)(293,295)(294,296)(301,303)(302,304)
(305,313)(306,314)(307,315)(308,316)(309,319)(310,320)(311,317)(312,318)
(321,353)(322,354)(323,355)(324,356)(325,359)(326,360)(327,357)(328,358)
(329,361)(330,362)(331,363)(332,364)(333,367)(334,368)(335,365)(336,366)
(337,377)(338,378)(339,379)(340,380)(341,383)(342,384)(343,381)(344,382)
(345,369)(346,370)(347,371)(348,372)(349,375)(350,376)(351,373)(352,374);;
s1 := (  1,161)(  2,162)(  3,165)(  4,166)(  5,163)(  6,164)(  7,167)(  8,168)
(  9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)
( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)
( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)
( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)
( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)
( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)
( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)
( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)
( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)
( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)
( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)
(193,353)(194,354)(195,357)(196,358)(197,355)(198,356)(199,359)(200,360)
(201,374)(202,373)(203,370)(204,369)(205,376)(206,375)(207,372)(208,371)
(209,364)(210,363)(211,368)(212,367)(213,362)(214,361)(215,366)(216,365)
(217,384)(218,383)(219,380)(220,379)(221,382)(222,381)(223,378)(224,377)
(225,321)(226,322)(227,325)(228,326)(229,323)(230,324)(231,327)(232,328)
(233,342)(234,341)(235,338)(236,337)(237,344)(238,343)(239,340)(240,339)
(241,332)(242,331)(243,336)(244,335)(245,330)(246,329)(247,334)(248,333)
(249,352)(250,351)(251,348)(252,347)(253,350)(254,349)(255,346)(256,345)
(257,289)(258,290)(259,293)(260,294)(261,291)(262,292)(263,295)(264,296)
(265,310)(266,309)(267,306)(268,305)(269,312)(270,311)(271,308)(272,307)
(273,300)(274,299)(275,304)(276,303)(277,298)(278,297)(279,302)(280,301)
(281,320)(282,319)(283,316)(284,315)(285,318)(286,317)(287,314)(288,313);;
s2 := (  1,201)(  2,202)(  3,203)(  4,204)(  5,208)(  6,207)(  7,206)(  8,205)
(  9,193)( 10,194)( 11,195)( 12,196)( 13,200)( 14,199)( 15,198)( 16,197)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,216)( 22,215)( 23,214)( 24,213)
( 25,217)( 26,218)( 27,219)( 28,220)( 29,224)( 30,223)( 31,222)( 32,221)
( 33,265)( 34,266)( 35,267)( 36,268)( 37,272)( 38,271)( 39,270)( 40,269)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,264)( 46,263)( 47,262)( 48,261)
( 49,273)( 50,274)( 51,275)( 52,276)( 53,280)( 54,279)( 55,278)( 56,277)
( 57,281)( 58,282)( 59,283)( 60,284)( 61,288)( 62,287)( 63,286)( 64,285)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,240)( 70,239)( 71,238)( 72,237)
( 73,225)( 74,226)( 75,227)( 76,228)( 77,232)( 78,231)( 79,230)( 80,229)
( 81,241)( 82,242)( 83,243)( 84,244)( 85,248)( 86,247)( 87,246)( 88,245)
( 89,249)( 90,250)( 91,251)( 92,252)( 93,256)( 94,255)( 95,254)( 96,253)
( 97,297)( 98,298)( 99,299)(100,300)(101,304)(102,303)(103,302)(104,301)
(105,289)(106,290)(107,291)(108,292)(109,296)(110,295)(111,294)(112,293)
(113,305)(114,306)(115,307)(116,308)(117,312)(118,311)(119,310)(120,309)
(121,313)(122,314)(123,315)(124,316)(125,320)(126,319)(127,318)(128,317)
(129,361)(130,362)(131,363)(132,364)(133,368)(134,367)(135,366)(136,365)
(137,353)(138,354)(139,355)(140,356)(141,360)(142,359)(143,358)(144,357)
(145,369)(146,370)(147,371)(148,372)(149,376)(150,375)(151,374)(152,373)
(153,377)(154,378)(155,379)(156,380)(157,384)(158,383)(159,382)(160,381)
(161,329)(162,330)(163,331)(164,332)(165,336)(166,335)(167,334)(168,333)
(169,321)(170,322)(171,323)(172,324)(173,328)(174,327)(175,326)(176,325)
(177,337)(178,338)(179,339)(180,340)(181,344)(182,343)(183,342)(184,341)
(185,345)(186,346)(187,347)(188,348)(189,352)(190,351)(191,350)(192,349);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope