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