Polytope of Type {4,3,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,3,6}*768
Also Known As : Dual of 4T4(2,0), {{4,3},{3,6}4}. if this polytope has another name.
Group : SmallGroup(768,1088539)
Rank : 4
Schlafli Type : {4,3,6}
Number of vertices, edges, etc : 16, 32, 48, 16
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Locally Toroidal
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 : {4,3,3}*384, {4,3,6}*384a, {4,3,6}*384b
4-fold quotients : {4,3,3}*192
8-fold quotients : {2,3,6}*96
16-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := ( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 16)( 6, 15)( 7, 14)( 8, 13)
( 21, 24)( 22, 23)( 29, 32)( 30, 31)( 33, 73)( 34, 74)( 35, 75)( 36, 76)
( 37, 80)( 38, 79)( 39, 78)( 40, 77)( 41, 65)( 42, 66)( 43, 67)( 44, 68)
( 45, 72)( 46, 71)( 47, 70)( 48, 69)( 49, 81)( 50, 82)( 51, 83)( 52, 84)
( 53, 88)( 54, 87)( 55, 86)( 56, 85)( 57, 89)( 58, 90)( 59, 91)( 60, 92)
( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 97,105)( 98,106)( 99,107)(100,108)
(101,112)(102,111)(103,110)(104,109)(117,120)(118,119)(125,128)(126,127)
(129,169)(130,170)(131,171)(132,172)(133,176)(134,175)(135,174)(136,173)
(137,161)(138,162)(139,163)(140,164)(141,168)(142,167)(143,166)(144,165)
(145,177)(146,178)(147,179)(148,180)(149,184)(150,183)(151,182)(152,181)
(153,185)(154,186)(155,187)(156,188)(157,192)(158,191)(159,190)(160,189)
(193,201)(194,202)(195,203)(196,204)(197,208)(198,207)(199,206)(200,205)
(213,216)(214,215)(221,224)(222,223)(225,265)(226,266)(227,267)(228,268)
(229,272)(230,271)(231,270)(232,269)(233,257)(234,258)(235,259)(236,260)
(237,264)(238,263)(239,262)(240,261)(241,273)(242,274)(243,275)(244,276)
(245,280)(246,279)(247,278)(248,277)(249,281)(250,282)(251,283)(252,284)
(253,288)(254,287)(255,286)(256,285)(289,297)(290,298)(291,299)(292,300)
(293,304)(294,303)(295,302)(296,301)(309,312)(310,311)(317,320)(318,319)
(321,361)(322,362)(323,363)(324,364)(325,368)(326,367)(327,366)(328,365)
(329,353)(330,354)(331,355)(332,356)(333,360)(334,359)(335,358)(336,357)
(337,369)(338,370)(339,371)(340,372)(341,376)(342,375)(343,374)(344,373)
(345,377)(346,378)(347,379)(348,380)(349,384)(350,383)(351,382)(352,381);;
s2 := ( 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);;
s3 := ( 1,193)( 2,194)( 3,195)( 4,196)( 5,199)( 6,200)( 7,197)( 8,198)
( 9,201)( 10,202)( 11,203)( 12,204)( 13,207)( 14,208)( 15,205)( 16,206)
( 17,217)( 18,218)( 19,219)( 20,220)( 21,223)( 22,224)( 23,221)( 24,222)
( 25,209)( 26,210)( 27,211)( 28,212)( 29,215)( 30,216)( 31,213)( 32,214)
( 33,257)( 34,258)( 35,259)( 36,260)( 37,263)( 38,264)( 39,261)( 40,262)
( 41,265)( 42,266)( 43,267)( 44,268)( 45,271)( 46,272)( 47,269)( 48,270)
( 49,281)( 50,282)( 51,283)( 52,284)( 53,287)( 54,288)( 55,285)( 56,286)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,279)( 62,280)( 63,277)( 64,278)
( 65,225)( 66,226)( 67,227)( 68,228)( 69,231)( 70,232)( 71,229)( 72,230)
( 73,233)( 74,234)( 75,235)( 76,236)( 77,239)( 78,240)( 79,237)( 80,238)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,255)( 86,256)( 87,253)( 88,254)
( 89,241)( 90,242)( 91,243)( 92,244)( 93,247)( 94,248)( 95,245)( 96,246)
( 97,289)( 98,290)( 99,291)(100,292)(101,295)(102,296)(103,293)(104,294)
(105,297)(106,298)(107,299)(108,300)(109,303)(110,304)(111,301)(112,302)
(113,313)(114,314)(115,315)(116,316)(117,319)(118,320)(119,317)(120,318)
(121,305)(122,306)(123,307)(124,308)(125,311)(126,312)(127,309)(128,310)
(129,353)(130,354)(131,355)(132,356)(133,359)(134,360)(135,357)(136,358)
(137,361)(138,362)(139,363)(140,364)(141,367)(142,368)(143,365)(144,366)
(145,377)(146,378)(147,379)(148,380)(149,383)(150,384)(151,381)(152,382)
(153,369)(154,370)(155,371)(156,372)(157,375)(158,376)(159,373)(160,374)
(161,321)(162,322)(163,323)(164,324)(165,327)(166,328)(167,325)(168,326)
(169,329)(170,330)(171,331)(172,332)(173,335)(174,336)(175,333)(176,334)
(177,345)(178,346)(179,347)(180,348)(181,351)(182,352)(183,349)(184,350)
(185,337)(186,338)(187,339)(188,340)(189,343)(190,344)(191,341)(192,342);;
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, s1*s2*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := 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);
s1 := Sym(384)!( 1, 9)( 2, 10)( 3, 11)( 4, 12)( 5, 16)( 6, 15)( 7, 14)
( 8, 13)( 21, 24)( 22, 23)( 29, 32)( 30, 31)( 33, 73)( 34, 74)( 35, 75)
( 36, 76)( 37, 80)( 38, 79)( 39, 78)( 40, 77)( 41, 65)( 42, 66)( 43, 67)
( 44, 68)( 45, 72)( 46, 71)( 47, 70)( 48, 69)( 49, 81)( 50, 82)( 51, 83)
( 52, 84)( 53, 88)( 54, 87)( 55, 86)( 56, 85)( 57, 89)( 58, 90)( 59, 91)
( 60, 92)( 61, 96)( 62, 95)( 63, 94)( 64, 93)( 97,105)( 98,106)( 99,107)
(100,108)(101,112)(102,111)(103,110)(104,109)(117,120)(118,119)(125,128)
(126,127)(129,169)(130,170)(131,171)(132,172)(133,176)(134,175)(135,174)
(136,173)(137,161)(138,162)(139,163)(140,164)(141,168)(142,167)(143,166)
(144,165)(145,177)(146,178)(147,179)(148,180)(149,184)(150,183)(151,182)
(152,181)(153,185)(154,186)(155,187)(156,188)(157,192)(158,191)(159,190)
(160,189)(193,201)(194,202)(195,203)(196,204)(197,208)(198,207)(199,206)
(200,205)(213,216)(214,215)(221,224)(222,223)(225,265)(226,266)(227,267)
(228,268)(229,272)(230,271)(231,270)(232,269)(233,257)(234,258)(235,259)
(236,260)(237,264)(238,263)(239,262)(240,261)(241,273)(242,274)(243,275)
(244,276)(245,280)(246,279)(247,278)(248,277)(249,281)(250,282)(251,283)
(252,284)(253,288)(254,287)(255,286)(256,285)(289,297)(290,298)(291,299)
(292,300)(293,304)(294,303)(295,302)(296,301)(309,312)(310,311)(317,320)
(318,319)(321,361)(322,362)(323,363)(324,364)(325,368)(326,367)(327,366)
(328,365)(329,353)(330,354)(331,355)(332,356)(333,360)(334,359)(335,358)
(336,357)(337,369)(338,370)(339,371)(340,372)(341,376)(342,375)(343,374)
(344,373)(345,377)(346,378)(347,379)(348,380)(349,384)(350,383)(351,382)
(352,381);
s2 := 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);
s3 := Sym(384)!( 1,193)( 2,194)( 3,195)( 4,196)( 5,199)( 6,200)( 7,197)
( 8,198)( 9,201)( 10,202)( 11,203)( 12,204)( 13,207)( 14,208)( 15,205)
( 16,206)( 17,217)( 18,218)( 19,219)( 20,220)( 21,223)( 22,224)( 23,221)
( 24,222)( 25,209)( 26,210)( 27,211)( 28,212)( 29,215)( 30,216)( 31,213)
( 32,214)( 33,257)( 34,258)( 35,259)( 36,260)( 37,263)( 38,264)( 39,261)
( 40,262)( 41,265)( 42,266)( 43,267)( 44,268)( 45,271)( 46,272)( 47,269)
( 48,270)( 49,281)( 50,282)( 51,283)( 52,284)( 53,287)( 54,288)( 55,285)
( 56,286)( 57,273)( 58,274)( 59,275)( 60,276)( 61,279)( 62,280)( 63,277)
( 64,278)( 65,225)( 66,226)( 67,227)( 68,228)( 69,231)( 70,232)( 71,229)
( 72,230)( 73,233)( 74,234)( 75,235)( 76,236)( 77,239)( 78,240)( 79,237)
( 80,238)( 81,249)( 82,250)( 83,251)( 84,252)( 85,255)( 86,256)( 87,253)
( 88,254)( 89,241)( 90,242)( 91,243)( 92,244)( 93,247)( 94,248)( 95,245)
( 96,246)( 97,289)( 98,290)( 99,291)(100,292)(101,295)(102,296)(103,293)
(104,294)(105,297)(106,298)(107,299)(108,300)(109,303)(110,304)(111,301)
(112,302)(113,313)(114,314)(115,315)(116,316)(117,319)(118,320)(119,317)
(120,318)(121,305)(122,306)(123,307)(124,308)(125,311)(126,312)(127,309)
(128,310)(129,353)(130,354)(131,355)(132,356)(133,359)(134,360)(135,357)
(136,358)(137,361)(138,362)(139,363)(140,364)(141,367)(142,368)(143,365)
(144,366)(145,377)(146,378)(147,379)(148,380)(149,383)(150,384)(151,381)
(152,382)(153,369)(154,370)(155,371)(156,372)(157,375)(158,376)(159,373)
(160,374)(161,321)(162,322)(163,323)(164,324)(165,327)(166,328)(167,325)
(168,326)(169,329)(170,330)(171,331)(172,332)(173,335)(174,336)(175,333)
(176,334)(177,345)(178,346)(179,347)(180,348)(181,351)(182,352)(183,349)
(184,350)(185,337)(186,338)(187,339)(188,340)(189,343)(190,344)(191,341)
(192,342);
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,
s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >;
References : - Theorem 11C7,11C8, McMullen P., Schulte, E.; Abstract Regular Polytopes (\
Cambridge University Press, 2002)
to this polytope