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Polytope of Type {6,8}

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

References : None.
to this polytope