Polytope of Type {6,6,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*768b
if this polytope has a name.
Group : SmallGroup(768,1088539)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 8, 48, 32, 8
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 4
Special Properties :
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,4}*384a, {6,3,4}*384a
4-fold quotients : {3,3,4}*192
16-fold quotients : {3,3,2}*48
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 := ( 1,257)( 2,258)( 3,261)( 4,262)( 5,259)( 6,260)( 7,263)( 8,264)
( 9,273)( 10,274)( 11,277)( 12,278)( 13,275)( 14,276)( 15,279)( 16,280)
( 17,265)( 18,266)( 19,269)( 20,270)( 21,267)( 22,268)( 23,271)( 24,272)
( 25,281)( 26,282)( 27,285)( 28,286)( 29,283)( 30,284)( 31,287)( 32,288)
( 33,225)( 34,226)( 35,229)( 36,230)( 37,227)( 38,228)( 39,231)( 40,232)
( 41,241)( 42,242)( 43,245)( 44,246)( 45,243)( 46,244)( 47,247)( 48,248)
( 49,233)( 50,234)( 51,237)( 52,238)( 53,235)( 54,236)( 55,239)( 56,240)
( 57,249)( 58,250)( 59,253)( 60,254)( 61,251)( 62,252)( 63,255)( 64,256)
( 65,193)( 66,194)( 67,197)( 68,198)( 69,195)( 70,196)( 71,199)( 72,200)
( 73,209)( 74,210)( 75,213)( 76,214)( 77,211)( 78,212)( 79,215)( 80,216)
( 81,201)( 82,202)( 83,205)( 84,206)( 85,203)( 86,204)( 87,207)( 88,208)
( 89,217)( 90,218)( 91,221)( 92,222)( 93,219)( 94,220)( 95,223)( 96,224)
( 97,353)( 98,354)( 99,357)(100,358)(101,355)(102,356)(103,359)(104,360)
(105,369)(106,370)(107,373)(108,374)(109,371)(110,372)(111,375)(112,376)
(113,361)(114,362)(115,365)(116,366)(117,363)(118,364)(119,367)(120,368)
(121,377)(122,378)(123,381)(124,382)(125,379)(126,380)(127,383)(128,384)
(129,321)(130,322)(131,325)(132,326)(133,323)(134,324)(135,327)(136,328)
(137,337)(138,338)(139,341)(140,342)(141,339)(142,340)(143,343)(144,344)
(145,329)(146,330)(147,333)(148,334)(149,331)(150,332)(151,335)(152,336)
(153,345)(154,346)(155,349)(156,350)(157,347)(158,348)(159,351)(160,352)
(161,289)(162,290)(163,293)(164,294)(165,291)(166,292)(167,295)(168,296)
(169,305)(170,306)(171,309)(172,310)(173,307)(174,308)(175,311)(176,312)
(177,297)(178,298)(179,301)(180,302)(181,299)(182,300)(183,303)(184,304)
(185,313)(186,314)(187,317)(188,318)(189,315)(190,316)(191,319)(192,320);;
s2 := ( 1, 11)( 2, 12)( 3, 9)( 4, 10)( 5, 14)( 6, 13)( 7, 16)( 8, 15)
( 17, 19)( 18, 20)( 21, 22)( 23, 24)( 25, 27)( 26, 28)( 29, 30)( 31, 32)
( 33, 75)( 34, 76)( 35, 73)( 36, 74)( 37, 78)( 38, 77)( 39, 80)( 40, 79)
( 41, 67)( 42, 68)( 43, 65)( 44, 66)( 45, 70)( 46, 69)( 47, 72)( 48, 71)
( 49, 83)( 50, 84)( 51, 81)( 52, 82)( 53, 86)( 54, 85)( 55, 88)( 56, 87)
( 57, 91)( 58, 92)( 59, 89)( 60, 90)( 61, 94)( 62, 93)( 63, 96)( 64, 95)
( 97,107)( 98,108)( 99,105)(100,106)(101,110)(102,109)(103,112)(104,111)
(113,115)(114,116)(117,118)(119,120)(121,123)(122,124)(125,126)(127,128)
(129,171)(130,172)(131,169)(132,170)(133,174)(134,173)(135,176)(136,175)
(137,163)(138,164)(139,161)(140,162)(141,166)(142,165)(143,168)(144,167)
(145,179)(146,180)(147,177)(148,178)(149,182)(150,181)(151,184)(152,183)
(153,187)(154,188)(155,185)(156,186)(157,190)(158,189)(159,192)(160,191)
(193,203)(194,204)(195,201)(196,202)(197,206)(198,205)(199,208)(200,207)
(209,211)(210,212)(213,214)(215,216)(217,219)(218,220)(221,222)(223,224)
(225,267)(226,268)(227,265)(228,266)(229,270)(230,269)(231,272)(232,271)
(233,259)(234,260)(235,257)(236,258)(237,262)(238,261)(239,264)(240,263)
(241,275)(242,276)(243,273)(244,274)(245,278)(246,277)(247,280)(248,279)
(249,283)(250,284)(251,281)(252,282)(253,286)(254,285)(255,288)(256,287)
(289,299)(290,300)(291,297)(292,298)(293,302)(294,301)(295,304)(296,303)
(305,307)(306,308)(309,310)(311,312)(313,315)(314,316)(317,318)(319,320)
(321,363)(322,364)(323,361)(324,362)(325,366)(326,365)(327,368)(328,367)
(329,355)(330,356)(331,353)(332,354)(333,358)(334,357)(335,360)(336,359)
(337,371)(338,372)(339,369)(340,370)(341,374)(342,373)(343,376)(344,375)
(345,379)(346,380)(347,377)(348,378)(349,382)(350,381)(351,384)(352,383);;
s3 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)( 16,109)
( 17,118)( 18,117)( 19,120)( 20,119)( 21,114)( 22,113)( 23,116)( 24,115)
( 25,128)( 26,127)( 27,126)( 28,125)( 29,124)( 30,123)( 31,122)( 32,121)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)( 48,141)
( 49,150)( 50,149)( 51,152)( 52,151)( 53,146)( 54,145)( 55,148)( 56,147)
( 57,160)( 58,159)( 59,158)( 60,157)( 61,156)( 62,155)( 63,154)( 64,153)
( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)( 72,168)
( 73,172)( 74,171)( 75,170)( 76,169)( 77,176)( 78,175)( 79,174)( 80,173)
( 81,182)( 82,181)( 83,184)( 84,183)( 85,178)( 86,177)( 87,180)( 88,179)
( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)( 96,185)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,300)(202,299)(203,298)(204,297)(205,304)(206,303)(207,302)(208,301)
(209,310)(210,309)(211,312)(212,311)(213,306)(214,305)(215,308)(216,307)
(217,320)(218,319)(219,318)(220,317)(221,316)(222,315)(223,314)(224,313)
(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)(232,328)
(233,332)(234,331)(235,330)(236,329)(237,336)(238,335)(239,334)(240,333)
(241,342)(242,341)(243,344)(244,343)(245,338)(246,337)(247,340)(248,339)
(249,352)(250,351)(251,350)(252,349)(253,348)(254,347)(255,346)(256,345)
(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)(264,360)
(265,364)(266,363)(267,362)(268,361)(269,368)(270,367)(271,366)(272,365)
(273,374)(274,373)(275,376)(276,375)(277,370)(278,369)(279,372)(280,371)
(281,384)(282,383)(283,382)(284,381)(285,380)(286,379)(287,378)(288,377);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s3*s1*s2*s3*s1*s2*s3*s0*s1*s2*s3*s2*s0*s1*s2 ];;
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)!( 1,257)( 2,258)( 3,261)( 4,262)( 5,259)( 6,260)( 7,263)
( 8,264)( 9,273)( 10,274)( 11,277)( 12,278)( 13,275)( 14,276)( 15,279)
( 16,280)( 17,265)( 18,266)( 19,269)( 20,270)( 21,267)( 22,268)( 23,271)
( 24,272)( 25,281)( 26,282)( 27,285)( 28,286)( 29,283)( 30,284)( 31,287)
( 32,288)( 33,225)( 34,226)( 35,229)( 36,230)( 37,227)( 38,228)( 39,231)
( 40,232)( 41,241)( 42,242)( 43,245)( 44,246)( 45,243)( 46,244)( 47,247)
( 48,248)( 49,233)( 50,234)( 51,237)( 52,238)( 53,235)( 54,236)( 55,239)
( 56,240)( 57,249)( 58,250)( 59,253)( 60,254)( 61,251)( 62,252)( 63,255)
( 64,256)( 65,193)( 66,194)( 67,197)( 68,198)( 69,195)( 70,196)( 71,199)
( 72,200)( 73,209)( 74,210)( 75,213)( 76,214)( 77,211)( 78,212)( 79,215)
( 80,216)( 81,201)( 82,202)( 83,205)( 84,206)( 85,203)( 86,204)( 87,207)
( 88,208)( 89,217)( 90,218)( 91,221)( 92,222)( 93,219)( 94,220)( 95,223)
( 96,224)( 97,353)( 98,354)( 99,357)(100,358)(101,355)(102,356)(103,359)
(104,360)(105,369)(106,370)(107,373)(108,374)(109,371)(110,372)(111,375)
(112,376)(113,361)(114,362)(115,365)(116,366)(117,363)(118,364)(119,367)
(120,368)(121,377)(122,378)(123,381)(124,382)(125,379)(126,380)(127,383)
(128,384)(129,321)(130,322)(131,325)(132,326)(133,323)(134,324)(135,327)
(136,328)(137,337)(138,338)(139,341)(140,342)(141,339)(142,340)(143,343)
(144,344)(145,329)(146,330)(147,333)(148,334)(149,331)(150,332)(151,335)
(152,336)(153,345)(154,346)(155,349)(156,350)(157,347)(158,348)(159,351)
(160,352)(161,289)(162,290)(163,293)(164,294)(165,291)(166,292)(167,295)
(168,296)(169,305)(170,306)(171,309)(172,310)(173,307)(174,308)(175,311)
(176,312)(177,297)(178,298)(179,301)(180,302)(181,299)(182,300)(183,303)
(184,304)(185,313)(186,314)(187,317)(188,318)(189,315)(190,316)(191,319)
(192,320);
s2 := Sym(384)!( 1, 11)( 2, 12)( 3, 9)( 4, 10)( 5, 14)( 6, 13)( 7, 16)
( 8, 15)( 17, 19)( 18, 20)( 21, 22)( 23, 24)( 25, 27)( 26, 28)( 29, 30)
( 31, 32)( 33, 75)( 34, 76)( 35, 73)( 36, 74)( 37, 78)( 38, 77)( 39, 80)
( 40, 79)( 41, 67)( 42, 68)( 43, 65)( 44, 66)( 45, 70)( 46, 69)( 47, 72)
( 48, 71)( 49, 83)( 50, 84)( 51, 81)( 52, 82)( 53, 86)( 54, 85)( 55, 88)
( 56, 87)( 57, 91)( 58, 92)( 59, 89)( 60, 90)( 61, 94)( 62, 93)( 63, 96)
( 64, 95)( 97,107)( 98,108)( 99,105)(100,106)(101,110)(102,109)(103,112)
(104,111)(113,115)(114,116)(117,118)(119,120)(121,123)(122,124)(125,126)
(127,128)(129,171)(130,172)(131,169)(132,170)(133,174)(134,173)(135,176)
(136,175)(137,163)(138,164)(139,161)(140,162)(141,166)(142,165)(143,168)
(144,167)(145,179)(146,180)(147,177)(148,178)(149,182)(150,181)(151,184)
(152,183)(153,187)(154,188)(155,185)(156,186)(157,190)(158,189)(159,192)
(160,191)(193,203)(194,204)(195,201)(196,202)(197,206)(198,205)(199,208)
(200,207)(209,211)(210,212)(213,214)(215,216)(217,219)(218,220)(221,222)
(223,224)(225,267)(226,268)(227,265)(228,266)(229,270)(230,269)(231,272)
(232,271)(233,259)(234,260)(235,257)(236,258)(237,262)(238,261)(239,264)
(240,263)(241,275)(242,276)(243,273)(244,274)(245,278)(246,277)(247,280)
(248,279)(249,283)(250,284)(251,281)(252,282)(253,286)(254,285)(255,288)
(256,287)(289,299)(290,300)(291,297)(292,298)(293,302)(294,301)(295,304)
(296,303)(305,307)(306,308)(309,310)(311,312)(313,315)(314,316)(317,318)
(319,320)(321,363)(322,364)(323,361)(324,362)(325,366)(326,365)(327,368)
(328,367)(329,355)(330,356)(331,353)(332,354)(333,358)(334,357)(335,360)
(336,359)(337,371)(338,372)(339,369)(340,370)(341,374)(342,373)(343,376)
(344,375)(345,379)(346,380)(347,377)(348,378)(349,382)(350,381)(351,384)
(352,383);
s3 := Sym(384)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,108)( 10,107)( 11,106)( 12,105)( 13,112)( 14,111)( 15,110)
( 16,109)( 17,118)( 18,117)( 19,120)( 20,119)( 21,114)( 22,113)( 23,116)
( 24,115)( 25,128)( 26,127)( 27,126)( 28,125)( 29,124)( 30,123)( 31,122)
( 32,121)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,140)( 42,139)( 43,138)( 44,137)( 45,144)( 46,143)( 47,142)
( 48,141)( 49,150)( 50,149)( 51,152)( 52,151)( 53,146)( 54,145)( 55,148)
( 56,147)( 57,160)( 58,159)( 59,158)( 60,157)( 61,156)( 62,155)( 63,154)
( 64,153)( 65,161)( 66,162)( 67,163)( 68,164)( 69,165)( 70,166)( 71,167)
( 72,168)( 73,172)( 74,171)( 75,170)( 76,169)( 77,176)( 78,175)( 79,174)
( 80,173)( 81,182)( 82,181)( 83,184)( 84,183)( 85,178)( 86,177)( 87,180)
( 88,179)( 89,192)( 90,191)( 91,190)( 92,189)( 93,188)( 94,187)( 95,186)
( 96,185)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,300)(202,299)(203,298)(204,297)(205,304)(206,303)(207,302)
(208,301)(209,310)(210,309)(211,312)(212,311)(213,306)(214,305)(215,308)
(216,307)(217,320)(218,319)(219,318)(220,317)(221,316)(222,315)(223,314)
(224,313)(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)
(232,328)(233,332)(234,331)(235,330)(236,329)(237,336)(238,335)(239,334)
(240,333)(241,342)(242,341)(243,344)(244,343)(245,338)(246,337)(247,340)
(248,339)(249,352)(250,351)(251,350)(252,349)(253,348)(254,347)(255,346)
(256,345)(257,353)(258,354)(259,355)(260,356)(261,357)(262,358)(263,359)
(264,360)(265,364)(266,363)(267,362)(268,361)(269,368)(270,367)(271,366)
(272,365)(273,374)(274,373)(275,376)(276,375)(277,370)(278,369)(279,372)
(280,371)(281,384)(282,383)(283,382)(284,381)(285,380)(286,379)(287,378)
(288,377);
poly := sub<Sym(384)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s0*s3*s1*s2*s3*s1*s2*s3*s0*s1*s2*s3*s2*s0*s1*s2 >;
References : None.
to this polytope