Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*768c
if this polytope has a name.
Group : SmallGroup(768,1086301)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 64, 192, 32
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {12,3}*384, {6,6}*384d
   4-fold quotients : {6,3}*192, {12,6}*192b
   8-fold quotients : {12,3}*96, {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 := (  3,  4)(  5,  6)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 17, 28)( 18, 27)
( 19, 25)( 20, 26)( 21, 31)( 22, 32)( 23, 30)( 24, 29)( 33, 65)( 34, 66)
( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)( 40, 72)( 41, 79)( 42, 80)
( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 73)( 48, 74)( 49, 92)( 50, 91)
( 51, 89)( 52, 90)( 53, 95)( 54, 96)( 55, 94)( 56, 93)( 57, 83)( 58, 84)
( 59, 82)( 60, 81)( 61, 88)( 62, 87)( 63, 85)( 64, 86)( 97,103)( 98,104)
( 99,102)(100,101)(107,108)(109,110)(113,126)(114,125)(115,127)(116,128)
(117,121)(118,122)(119,124)(120,123)(129,167)(130,168)(131,166)(132,165)
(133,164)(134,163)(135,161)(136,162)(137,169)(138,170)(139,172)(140,171)
(141,174)(142,173)(143,175)(144,176)(145,190)(146,189)(147,191)(148,192)
(149,185)(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)(156,183)
(157,178)(158,177)(159,179)(160,180)(195,196)(197,198)(201,207)(202,208)
(203,206)(204,205)(209,220)(210,219)(211,217)(212,218)(213,223)(214,224)
(215,222)(216,221)(225,257)(226,258)(227,260)(228,259)(229,262)(230,261)
(231,263)(232,264)(233,271)(234,272)(235,270)(236,269)(237,268)(238,267)
(239,265)(240,266)(241,284)(242,283)(243,281)(244,282)(245,287)(246,288)
(247,286)(248,285)(249,275)(250,276)(251,274)(252,273)(253,280)(254,279)
(255,277)(256,278)(289,295)(290,296)(291,294)(292,293)(299,300)(301,302)
(305,318)(306,317)(307,319)(308,320)(309,313)(310,314)(311,316)(312,315)
(321,359)(322,360)(323,358)(324,357)(325,356)(326,355)(327,353)(328,354)
(329,361)(330,362)(331,364)(332,363)(333,366)(334,365)(335,367)(336,368)
(337,382)(338,381)(339,383)(340,384)(341,377)(342,378)(343,380)(344,379)
(345,373)(346,374)(347,376)(348,375)(349,370)(350,369)(351,371)(352,372);;
s1 := (  1,192)(  2,190)(  3,191)(  4,189)(  5,185)(  6,187)(  7,186)(  8,188)
(  9,175)( 10,173)( 11,176)( 12,174)( 13,170)( 14,172)( 15,169)( 16,171)
( 17,180)( 18,178)( 19,179)( 20,177)( 21,181)( 22,183)( 23,182)( 24,184)
( 25,165)( 26,167)( 27,166)( 28,168)( 29,164)( 30,162)( 31,163)( 32,161)
( 33,160)( 34,158)( 35,159)( 36,157)( 37,153)( 38,155)( 39,154)( 40,156)
( 41,143)( 42,141)( 43,144)( 44,142)( 45,138)( 46,140)( 47,137)( 48,139)
( 49,148)( 50,146)( 51,147)( 52,145)( 53,149)( 54,151)( 55,150)( 56,152)
( 57,133)( 58,135)( 59,134)( 60,136)( 61,132)( 62,130)( 63,131)( 64,129)
( 65,128)( 66,126)( 67,127)( 68,125)( 69,121)( 70,123)( 71,122)( 72,124)
( 73,111)( 74,109)( 75,112)( 76,110)( 77,106)( 78,108)( 79,105)( 80,107)
( 81,116)( 82,114)( 83,115)( 84,113)( 85,117)( 86,119)( 87,118)( 88,120)
( 89,101)( 90,103)( 91,102)( 92,104)( 93,100)( 94, 98)( 95, 99)( 96, 97)
(193,384)(194,382)(195,383)(196,381)(197,377)(198,379)(199,378)(200,380)
(201,367)(202,365)(203,368)(204,366)(205,362)(206,364)(207,361)(208,363)
(209,372)(210,370)(211,371)(212,369)(213,373)(214,375)(215,374)(216,376)
(217,357)(218,359)(219,358)(220,360)(221,356)(222,354)(223,355)(224,353)
(225,352)(226,350)(227,351)(228,349)(229,345)(230,347)(231,346)(232,348)
(233,335)(234,333)(235,336)(236,334)(237,330)(238,332)(239,329)(240,331)
(241,340)(242,338)(243,339)(244,337)(245,341)(246,343)(247,342)(248,344)
(249,325)(250,327)(251,326)(252,328)(253,324)(254,322)(255,323)(256,321)
(257,320)(258,318)(259,319)(260,317)(261,313)(262,315)(263,314)(264,316)
(265,303)(266,301)(267,304)(268,302)(269,298)(270,300)(271,297)(272,299)
(273,308)(274,306)(275,307)(276,305)(277,309)(278,311)(279,310)(280,312)
(281,293)(282,295)(283,294)(284,296)(285,292)(286,290)(287,291)(288,289);;
s2 := (  1,298)(  2,297)(  3,299)(  4,300)(  5,301)(  6,302)(  7,304)(  8,303)
(  9,290)( 10,289)( 11,291)( 12,292)( 13,293)( 14,294)( 15,296)( 16,295)
( 17,311)( 18,312)( 19,310)( 20,309)( 21,308)( 22,307)( 23,305)( 24,306)
( 25,314)( 26,313)( 27,315)( 28,316)( 29,317)( 30,318)( 31,320)( 32,319)
( 33,362)( 34,361)( 35,363)( 36,364)( 37,365)( 38,366)( 39,368)( 40,367)
( 41,354)( 42,353)( 43,355)( 44,356)( 45,357)( 46,358)( 47,360)( 48,359)
( 49,375)( 50,376)( 51,374)( 52,373)( 53,372)( 54,371)( 55,369)( 56,370)
( 57,378)( 58,377)( 59,379)( 60,380)( 61,381)( 62,382)( 63,384)( 64,383)
( 65,330)( 66,329)( 67,331)( 68,332)( 69,333)( 70,334)( 71,336)( 72,335)
( 73,322)( 74,321)( 75,323)( 76,324)( 77,325)( 78,326)( 79,328)( 80,327)
( 81,343)( 82,344)( 83,342)( 84,341)( 85,340)( 86,339)( 87,337)( 88,338)
( 89,346)( 90,345)( 91,347)( 92,348)( 93,349)( 94,350)( 95,352)( 96,351)
( 97,202)( 98,201)( 99,203)(100,204)(101,205)(102,206)(103,208)(104,207)
(105,194)(106,193)(107,195)(108,196)(109,197)(110,198)(111,200)(112,199)
(113,215)(114,216)(115,214)(116,213)(117,212)(118,211)(119,209)(120,210)
(121,218)(122,217)(123,219)(124,220)(125,221)(126,222)(127,224)(128,223)
(129,266)(130,265)(131,267)(132,268)(133,269)(134,270)(135,272)(136,271)
(137,258)(138,257)(139,259)(140,260)(141,261)(142,262)(143,264)(144,263)
(145,279)(146,280)(147,278)(148,277)(149,276)(150,275)(151,273)(152,274)
(153,282)(154,281)(155,283)(156,284)(157,285)(158,286)(159,288)(160,287)
(161,234)(162,233)(163,235)(164,236)(165,237)(166,238)(167,240)(168,239)
(169,226)(170,225)(171,227)(172,228)(173,229)(174,230)(175,232)(176,231)
(177,247)(178,248)(179,246)(180,245)(181,244)(182,243)(183,241)(184,242)
(185,250)(186,249)(187,251)(188,252)(189,253)(190,254)(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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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)!(  3,  4)(  5,  6)(  9, 15)( 10, 16)( 11, 14)( 12, 13)( 17, 28)
( 18, 27)( 19, 25)( 20, 26)( 21, 31)( 22, 32)( 23, 30)( 24, 29)( 33, 65)
( 34, 66)( 35, 68)( 36, 67)( 37, 70)( 38, 69)( 39, 71)( 40, 72)( 41, 79)
( 42, 80)( 43, 78)( 44, 77)( 45, 76)( 46, 75)( 47, 73)( 48, 74)( 49, 92)
( 50, 91)( 51, 89)( 52, 90)( 53, 95)( 54, 96)( 55, 94)( 56, 93)( 57, 83)
( 58, 84)( 59, 82)( 60, 81)( 61, 88)( 62, 87)( 63, 85)( 64, 86)( 97,103)
( 98,104)( 99,102)(100,101)(107,108)(109,110)(113,126)(114,125)(115,127)
(116,128)(117,121)(118,122)(119,124)(120,123)(129,167)(130,168)(131,166)
(132,165)(133,164)(134,163)(135,161)(136,162)(137,169)(138,170)(139,172)
(140,171)(141,174)(142,173)(143,175)(144,176)(145,190)(146,189)(147,191)
(148,192)(149,185)(150,186)(151,188)(152,187)(153,181)(154,182)(155,184)
(156,183)(157,178)(158,177)(159,179)(160,180)(195,196)(197,198)(201,207)
(202,208)(203,206)(204,205)(209,220)(210,219)(211,217)(212,218)(213,223)
(214,224)(215,222)(216,221)(225,257)(226,258)(227,260)(228,259)(229,262)
(230,261)(231,263)(232,264)(233,271)(234,272)(235,270)(236,269)(237,268)
(238,267)(239,265)(240,266)(241,284)(242,283)(243,281)(244,282)(245,287)
(246,288)(247,286)(248,285)(249,275)(250,276)(251,274)(252,273)(253,280)
(254,279)(255,277)(256,278)(289,295)(290,296)(291,294)(292,293)(299,300)
(301,302)(305,318)(306,317)(307,319)(308,320)(309,313)(310,314)(311,316)
(312,315)(321,359)(322,360)(323,358)(324,357)(325,356)(326,355)(327,353)
(328,354)(329,361)(330,362)(331,364)(332,363)(333,366)(334,365)(335,367)
(336,368)(337,382)(338,381)(339,383)(340,384)(341,377)(342,378)(343,380)
(344,379)(345,373)(346,374)(347,376)(348,375)(349,370)(350,369)(351,371)
(352,372);
s1 := Sym(384)!(  1,192)(  2,190)(  3,191)(  4,189)(  5,185)(  6,187)(  7,186)
(  8,188)(  9,175)( 10,173)( 11,176)( 12,174)( 13,170)( 14,172)( 15,169)
( 16,171)( 17,180)( 18,178)( 19,179)( 20,177)( 21,181)( 22,183)( 23,182)
( 24,184)( 25,165)( 26,167)( 27,166)( 28,168)( 29,164)( 30,162)( 31,163)
( 32,161)( 33,160)( 34,158)( 35,159)( 36,157)( 37,153)( 38,155)( 39,154)
( 40,156)( 41,143)( 42,141)( 43,144)( 44,142)( 45,138)( 46,140)( 47,137)
( 48,139)( 49,148)( 50,146)( 51,147)( 52,145)( 53,149)( 54,151)( 55,150)
( 56,152)( 57,133)( 58,135)( 59,134)( 60,136)( 61,132)( 62,130)( 63,131)
( 64,129)( 65,128)( 66,126)( 67,127)( 68,125)( 69,121)( 70,123)( 71,122)
( 72,124)( 73,111)( 74,109)( 75,112)( 76,110)( 77,106)( 78,108)( 79,105)
( 80,107)( 81,116)( 82,114)( 83,115)( 84,113)( 85,117)( 86,119)( 87,118)
( 88,120)( 89,101)( 90,103)( 91,102)( 92,104)( 93,100)( 94, 98)( 95, 99)
( 96, 97)(193,384)(194,382)(195,383)(196,381)(197,377)(198,379)(199,378)
(200,380)(201,367)(202,365)(203,368)(204,366)(205,362)(206,364)(207,361)
(208,363)(209,372)(210,370)(211,371)(212,369)(213,373)(214,375)(215,374)
(216,376)(217,357)(218,359)(219,358)(220,360)(221,356)(222,354)(223,355)
(224,353)(225,352)(226,350)(227,351)(228,349)(229,345)(230,347)(231,346)
(232,348)(233,335)(234,333)(235,336)(236,334)(237,330)(238,332)(239,329)
(240,331)(241,340)(242,338)(243,339)(244,337)(245,341)(246,343)(247,342)
(248,344)(249,325)(250,327)(251,326)(252,328)(253,324)(254,322)(255,323)
(256,321)(257,320)(258,318)(259,319)(260,317)(261,313)(262,315)(263,314)
(264,316)(265,303)(266,301)(267,304)(268,302)(269,298)(270,300)(271,297)
(272,299)(273,308)(274,306)(275,307)(276,305)(277,309)(278,311)(279,310)
(280,312)(281,293)(282,295)(283,294)(284,296)(285,292)(286,290)(287,291)
(288,289);
s2 := Sym(384)!(  1,298)(  2,297)(  3,299)(  4,300)(  5,301)(  6,302)(  7,304)
(  8,303)(  9,290)( 10,289)( 11,291)( 12,292)( 13,293)( 14,294)( 15,296)
( 16,295)( 17,311)( 18,312)( 19,310)( 20,309)( 21,308)( 22,307)( 23,305)
( 24,306)( 25,314)( 26,313)( 27,315)( 28,316)( 29,317)( 30,318)( 31,320)
( 32,319)( 33,362)( 34,361)( 35,363)( 36,364)( 37,365)( 38,366)( 39,368)
( 40,367)( 41,354)( 42,353)( 43,355)( 44,356)( 45,357)( 46,358)( 47,360)
( 48,359)( 49,375)( 50,376)( 51,374)( 52,373)( 53,372)( 54,371)( 55,369)
( 56,370)( 57,378)( 58,377)( 59,379)( 60,380)( 61,381)( 62,382)( 63,384)
( 64,383)( 65,330)( 66,329)( 67,331)( 68,332)( 69,333)( 70,334)( 71,336)
( 72,335)( 73,322)( 74,321)( 75,323)( 76,324)( 77,325)( 78,326)( 79,328)
( 80,327)( 81,343)( 82,344)( 83,342)( 84,341)( 85,340)( 86,339)( 87,337)
( 88,338)( 89,346)( 90,345)( 91,347)( 92,348)( 93,349)( 94,350)( 95,352)
( 96,351)( 97,202)( 98,201)( 99,203)(100,204)(101,205)(102,206)(103,208)
(104,207)(105,194)(106,193)(107,195)(108,196)(109,197)(110,198)(111,200)
(112,199)(113,215)(114,216)(115,214)(116,213)(117,212)(118,211)(119,209)
(120,210)(121,218)(122,217)(123,219)(124,220)(125,221)(126,222)(127,224)
(128,223)(129,266)(130,265)(131,267)(132,268)(133,269)(134,270)(135,272)
(136,271)(137,258)(138,257)(139,259)(140,260)(141,261)(142,262)(143,264)
(144,263)(145,279)(146,280)(147,278)(148,277)(149,276)(150,275)(151,273)
(152,274)(153,282)(154,281)(155,283)(156,284)(157,285)(158,286)(159,288)
(160,287)(161,234)(162,233)(163,235)(164,236)(165,237)(166,238)(167,240)
(168,239)(169,226)(170,225)(171,227)(172,228)(173,229)(174,230)(175,232)
(176,231)(177,247)(178,248)(179,246)(180,245)(181,244)(182,243)(183,241)
(184,242)(185,250)(186,249)(187,251)(188,252)(189,253)(190,254)(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, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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