Polytope of Type {6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8}*768m
if this polytope has a name.
Group : SmallGroup(768,1088539)
Rank : 3
Schlafli Type : {6,8}
Number of vertices, edges, etc : 48, 192, 64
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
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 : {6,8}*384b, {6,8}*384c, {6,4}*384a
4-fold quotients : {6,4}*192a
8-fold quotients : {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 := ( 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)( 5,206)( 6,205)( 7,208)( 8,207)
( 9,193)( 10,194)( 11,195)( 12,196)( 13,198)( 14,197)( 15,200)( 16,199)
( 17,217)( 18,218)( 19,219)( 20,220)( 21,222)( 22,221)( 23,224)( 24,223)
( 25,209)( 26,210)( 27,211)( 28,212)( 29,214)( 30,213)( 31,216)( 32,215)
( 33,233)( 34,234)( 35,235)( 36,236)( 37,238)( 38,237)( 39,240)( 40,239)
( 41,225)( 42,226)( 43,227)( 44,228)( 45,230)( 46,229)( 47,232)( 48,231)
( 49,249)( 50,250)( 51,251)( 52,252)( 53,254)( 54,253)( 55,256)( 56,255)
( 57,241)( 58,242)( 59,243)( 60,244)( 61,246)( 62,245)( 63,248)( 64,247)
( 65,265)( 66,266)( 67,267)( 68,268)( 69,270)( 70,269)( 71,272)( 72,271)
( 73,257)( 74,258)( 75,259)( 76,260)( 77,262)( 78,261)( 79,264)( 80,263)
( 81,281)( 82,282)( 83,283)( 84,284)( 85,286)( 86,285)( 87,288)( 88,287)
( 89,273)( 90,274)( 91,275)( 92,276)( 93,278)( 94,277)( 95,280)( 96,279)
( 97,297)( 98,298)( 99,299)(100,300)(101,302)(102,301)(103,304)(104,303)
(105,289)(106,290)(107,291)(108,292)(109,294)(110,293)(111,296)(112,295)
(113,313)(114,314)(115,315)(116,316)(117,318)(118,317)(119,320)(120,319)
(121,305)(122,306)(123,307)(124,308)(125,310)(126,309)(127,312)(128,311)
(129,329)(130,330)(131,331)(132,332)(133,334)(134,333)(135,336)(136,335)
(137,321)(138,322)(139,323)(140,324)(141,326)(142,325)(143,328)(144,327)
(145,345)(146,346)(147,347)(148,348)(149,350)(150,349)(151,352)(152,351)
(153,337)(154,338)(155,339)(156,340)(157,342)(158,341)(159,344)(160,343)
(161,361)(162,362)(163,363)(164,364)(165,366)(166,365)(167,368)(168,367)
(169,353)(170,354)(171,355)(172,356)(173,358)(174,357)(175,360)(176,359)
(177,377)(178,378)(179,379)(180,380)(181,382)(182,381)(183,384)(184,383)
(185,369)(186,370)(187,371)(188,372)(189,374)(190,373)(191,376)(192,375);;
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*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)( 5,206)( 6,205)( 7,208)
( 8,207)( 9,193)( 10,194)( 11,195)( 12,196)( 13,198)( 14,197)( 15,200)
( 16,199)( 17,217)( 18,218)( 19,219)( 20,220)( 21,222)( 22,221)( 23,224)
( 24,223)( 25,209)( 26,210)( 27,211)( 28,212)( 29,214)( 30,213)( 31,216)
( 32,215)( 33,233)( 34,234)( 35,235)( 36,236)( 37,238)( 38,237)( 39,240)
( 40,239)( 41,225)( 42,226)( 43,227)( 44,228)( 45,230)( 46,229)( 47,232)
( 48,231)( 49,249)( 50,250)( 51,251)( 52,252)( 53,254)( 54,253)( 55,256)
( 56,255)( 57,241)( 58,242)( 59,243)( 60,244)( 61,246)( 62,245)( 63,248)
( 64,247)( 65,265)( 66,266)( 67,267)( 68,268)( 69,270)( 70,269)( 71,272)
( 72,271)( 73,257)( 74,258)( 75,259)( 76,260)( 77,262)( 78,261)( 79,264)
( 80,263)( 81,281)( 82,282)( 83,283)( 84,284)( 85,286)( 86,285)( 87,288)
( 88,287)( 89,273)( 90,274)( 91,275)( 92,276)( 93,278)( 94,277)( 95,280)
( 96,279)( 97,297)( 98,298)( 99,299)(100,300)(101,302)(102,301)(103,304)
(104,303)(105,289)(106,290)(107,291)(108,292)(109,294)(110,293)(111,296)
(112,295)(113,313)(114,314)(115,315)(116,316)(117,318)(118,317)(119,320)
(120,319)(121,305)(122,306)(123,307)(124,308)(125,310)(126,309)(127,312)
(128,311)(129,329)(130,330)(131,331)(132,332)(133,334)(134,333)(135,336)
(136,335)(137,321)(138,322)(139,323)(140,324)(141,326)(142,325)(143,328)
(144,327)(145,345)(146,346)(147,347)(148,348)(149,350)(150,349)(151,352)
(152,351)(153,337)(154,338)(155,339)(156,340)(157,342)(158,341)(159,344)
(160,343)(161,361)(162,362)(163,363)(164,364)(165,366)(166,365)(167,368)
(168,367)(169,353)(170,354)(171,355)(172,356)(173,358)(174,357)(175,360)
(176,359)(177,377)(178,378)(179,379)(180,380)(181,382)(182,381)(183,384)
(184,383)(185,369)(186,370)(187,371)(188,372)(189,374)(190,373)(191,376)
(192,375);
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*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope