Polytope of Type {6,4,16}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,16}*768a
Also Known As : {{6,4|2},{4,16|2}}. if this polytope has another name.
Group : SmallGroup(768,323301)
Rank : 4
Schlafli Type : {6,4,16}
Number of vertices, edges, etc : 6, 12, 32, 16
Order of s0s1s2s3 : 48
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
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 : {6,4,8}*384a, {6,2,16}*384
3-fold quotients : {2,4,16}*256a
4-fold quotients : {3,2,16}*192, {6,4,4}*192, {6,2,8}*192
6-fold quotients : {2,4,8}*128a, {2,2,16}*128
8-fold quotients : {3,2,8}*96, {6,2,4}*96, {6,4,2}*96a
12-fold quotients : {2,4,4}*64, {2,2,8}*64
16-fold quotients : {3,2,4}*48, {6,2,2}*48
24-fold quotients : {2,2,4}*32, {2,4,2}*32
32-fold quotients : {3,2,2}*24
48-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)( 8, 57)
( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)( 16, 64)
( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)( 24, 71)
( 25, 73)( 26, 75)( 27, 74)( 28, 76)( 29, 78)( 30, 77)( 31, 79)( 32, 81)
( 33, 80)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 87)( 39, 86)( 40, 88)
( 41, 90)( 42, 89)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 96)( 48, 95)
( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)(104,153)
(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)(112,160)
(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)(120,167)
(121,169)(122,171)(123,170)(124,172)(125,174)(126,173)(127,175)(128,177)
(129,176)(130,178)(131,180)(132,179)(133,181)(134,183)(135,182)(136,184)
(137,186)(138,185)(139,187)(140,189)(141,188)(142,190)(143,192)(144,191)
(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)(200,249)
(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)(208,256)
(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)(216,263)
(217,265)(218,267)(219,266)(220,268)(221,270)(222,269)(223,271)(224,273)
(225,272)(226,274)(227,276)(228,275)(229,277)(230,279)(231,278)(232,280)
(233,282)(234,281)(235,283)(236,285)(237,284)(238,286)(239,288)(240,287)
(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)(296,345)
(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)(304,352)
(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)(312,359)
(313,361)(314,363)(315,362)(316,364)(317,366)(318,365)(319,367)(320,369)
(321,368)(322,370)(323,372)(324,371)(325,373)(326,375)(327,374)(328,376)
(329,378)(330,377)(331,379)(332,381)(333,380)(334,382)(335,384)(336,383);;
s1 := ( 1,147)( 2,146)( 3,145)( 4,150)( 5,149)( 6,148)( 7,153)( 8,152)
( 9,151)( 10,156)( 11,155)( 12,154)( 13,159)( 14,158)( 15,157)( 16,162)
( 17,161)( 18,160)( 19,165)( 20,164)( 21,163)( 22,168)( 23,167)( 24,166)
( 25,171)( 26,170)( 27,169)( 28,174)( 29,173)( 30,172)( 31,177)( 32,176)
( 33,175)( 34,180)( 35,179)( 36,178)( 37,183)( 38,182)( 39,181)( 40,186)
( 41,185)( 42,184)( 43,189)( 44,188)( 45,187)( 46,192)( 47,191)( 48,190)
( 49, 99)( 50, 98)( 51, 97)( 52,102)( 53,101)( 54,100)( 55,105)( 56,104)
( 57,103)( 58,108)( 59,107)( 60,106)( 61,111)( 62,110)( 63,109)( 64,114)
( 65,113)( 66,112)( 67,117)( 68,116)( 69,115)( 70,120)( 71,119)( 72,118)
( 73,123)( 74,122)( 75,121)( 76,126)( 77,125)( 78,124)( 79,129)( 80,128)
( 81,127)( 82,132)( 83,131)( 84,130)( 85,135)( 86,134)( 87,133)( 88,138)
( 89,137)( 90,136)( 91,141)( 92,140)( 93,139)( 94,144)( 95,143)( 96,142)
(193,339)(194,338)(195,337)(196,342)(197,341)(198,340)(199,345)(200,344)
(201,343)(202,348)(203,347)(204,346)(205,351)(206,350)(207,349)(208,354)
(209,353)(210,352)(211,357)(212,356)(213,355)(214,360)(215,359)(216,358)
(217,363)(218,362)(219,361)(220,366)(221,365)(222,364)(223,369)(224,368)
(225,367)(226,372)(227,371)(228,370)(229,375)(230,374)(231,373)(232,378)
(233,377)(234,376)(235,381)(236,380)(237,379)(238,384)(239,383)(240,382)
(241,291)(242,290)(243,289)(244,294)(245,293)(246,292)(247,297)(248,296)
(249,295)(250,300)(251,299)(252,298)(253,303)(254,302)(255,301)(256,306)
(257,305)(258,304)(259,309)(260,308)(261,307)(262,312)(263,311)(264,310)
(265,315)(266,314)(267,313)(268,318)(269,317)(270,316)(271,321)(272,320)
(273,319)(274,324)(275,323)(276,322)(277,327)(278,326)(279,325)(280,330)
(281,329)(282,328)(283,333)(284,332)(285,331)(286,336)(287,335)(288,334);;
s2 := ( 7, 10)( 8, 11)( 9, 12)( 19, 22)( 20, 23)( 21, 24)( 25, 31)( 26, 32)
( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)( 38, 44)( 39, 45)( 40, 46)
( 41, 47)( 42, 48)( 55, 58)( 56, 59)( 57, 60)( 67, 70)( 68, 71)( 69, 72)
( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 91)( 86, 92)
( 87, 93)( 88, 94)( 89, 95)( 90, 96)( 97,109)( 98,110)( 99,111)(100,112)
(101,113)(102,114)(103,118)(104,119)(105,120)(106,115)(107,116)(108,117)
(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)(128,134)
(129,135)(130,136)(131,137)(132,138)(145,157)(146,158)(147,159)(148,160)
(149,161)(150,162)(151,166)(152,167)(153,168)(154,163)(155,164)(156,165)
(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)
(177,183)(178,184)(179,185)(180,186)(193,217)(194,218)(195,219)(196,220)
(197,221)(198,222)(199,226)(200,227)(201,228)(202,223)(203,224)(204,225)
(205,229)(206,230)(207,231)(208,232)(209,233)(210,234)(211,238)(212,239)
(213,240)(214,235)(215,236)(216,237)(241,265)(242,266)(243,267)(244,268)
(245,269)(246,270)(247,274)(248,275)(249,276)(250,271)(251,272)(252,273)
(253,277)(254,278)(255,279)(256,280)(257,281)(258,282)(259,286)(260,287)
(261,288)(262,283)(263,284)(264,285)(289,325)(290,326)(291,327)(292,328)
(293,329)(294,330)(295,334)(296,335)(297,336)(298,331)(299,332)(300,333)
(301,313)(302,314)(303,315)(304,316)(305,317)(306,318)(307,322)(308,323)
(309,324)(310,319)(311,320)(312,321)(337,373)(338,374)(339,375)(340,376)
(341,377)(342,378)(343,382)(344,383)(345,384)(346,379)(347,380)(348,381)
(349,361)(350,362)(351,363)(352,364)(353,365)(354,366)(355,370)(356,371)
(357,372)(358,367)(359,368)(360,369);;
s3 := ( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,202)( 8,203)
( 9,204)( 10,199)( 11,200)( 12,201)( 13,205)( 14,206)( 15,207)( 16,208)
( 17,209)( 18,210)( 19,214)( 20,215)( 21,216)( 22,211)( 23,212)( 24,213)
( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)( 32,218)
( 33,219)( 34,220)( 35,221)( 36,222)( 37,235)( 38,236)( 39,237)( 40,238)
( 41,239)( 42,240)( 43,229)( 44,230)( 45,231)( 46,232)( 47,233)( 48,234)
( 49,241)( 50,242)( 51,243)( 52,244)( 53,245)( 54,246)( 55,250)( 56,251)
( 57,252)( 58,247)( 59,248)( 60,249)( 61,253)( 62,254)( 63,255)( 64,256)
( 65,257)( 66,258)( 67,262)( 68,263)( 69,264)( 70,259)( 71,260)( 72,261)
( 73,271)( 74,272)( 75,273)( 76,274)( 77,275)( 78,276)( 79,265)( 80,266)
( 81,267)( 82,268)( 83,269)( 84,270)( 85,283)( 86,284)( 87,285)( 88,286)
( 89,287)( 90,288)( 91,277)( 92,278)( 93,279)( 94,280)( 95,281)( 96,282)
( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,298)(104,299)
(105,300)(106,295)(107,296)(108,297)(109,301)(110,302)(111,303)(112,304)
(113,305)(114,306)(115,310)(116,311)(117,312)(118,307)(119,308)(120,309)
(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)(128,314)
(129,315)(130,316)(131,317)(132,318)(133,331)(134,332)(135,333)(136,334)
(137,335)(138,336)(139,325)(140,326)(141,327)(142,328)(143,329)(144,330)
(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,346)(152,347)
(153,348)(154,343)(155,344)(156,345)(157,349)(158,350)(159,351)(160,352)
(161,353)(162,354)(163,358)(164,359)(165,360)(166,355)(167,356)(168,357)
(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,361)(176,362)
(177,363)(178,364)(179,365)(180,366)(181,379)(182,380)(183,381)(184,382)
(185,383)(186,384)(187,373)(188,374)(189,375)(190,376)(191,377)(192,378);;
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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1, 49)( 2, 51)( 3, 50)( 4, 52)( 5, 54)( 6, 53)( 7, 55)
( 8, 57)( 9, 56)( 10, 58)( 11, 60)( 12, 59)( 13, 61)( 14, 63)( 15, 62)
( 16, 64)( 17, 66)( 18, 65)( 19, 67)( 20, 69)( 21, 68)( 22, 70)( 23, 72)
( 24, 71)( 25, 73)( 26, 75)( 27, 74)( 28, 76)( 29, 78)( 30, 77)( 31, 79)
( 32, 81)( 33, 80)( 34, 82)( 35, 84)( 36, 83)( 37, 85)( 38, 87)( 39, 86)
( 40, 88)( 41, 90)( 42, 89)( 43, 91)( 44, 93)( 45, 92)( 46, 94)( 47, 96)
( 48, 95)( 97,145)( 98,147)( 99,146)(100,148)(101,150)(102,149)(103,151)
(104,153)(105,152)(106,154)(107,156)(108,155)(109,157)(110,159)(111,158)
(112,160)(113,162)(114,161)(115,163)(116,165)(117,164)(118,166)(119,168)
(120,167)(121,169)(122,171)(123,170)(124,172)(125,174)(126,173)(127,175)
(128,177)(129,176)(130,178)(131,180)(132,179)(133,181)(134,183)(135,182)
(136,184)(137,186)(138,185)(139,187)(140,189)(141,188)(142,190)(143,192)
(144,191)(193,241)(194,243)(195,242)(196,244)(197,246)(198,245)(199,247)
(200,249)(201,248)(202,250)(203,252)(204,251)(205,253)(206,255)(207,254)
(208,256)(209,258)(210,257)(211,259)(212,261)(213,260)(214,262)(215,264)
(216,263)(217,265)(218,267)(219,266)(220,268)(221,270)(222,269)(223,271)
(224,273)(225,272)(226,274)(227,276)(228,275)(229,277)(230,279)(231,278)
(232,280)(233,282)(234,281)(235,283)(236,285)(237,284)(238,286)(239,288)
(240,287)(289,337)(290,339)(291,338)(292,340)(293,342)(294,341)(295,343)
(296,345)(297,344)(298,346)(299,348)(300,347)(301,349)(302,351)(303,350)
(304,352)(305,354)(306,353)(307,355)(308,357)(309,356)(310,358)(311,360)
(312,359)(313,361)(314,363)(315,362)(316,364)(317,366)(318,365)(319,367)
(320,369)(321,368)(322,370)(323,372)(324,371)(325,373)(326,375)(327,374)
(328,376)(329,378)(330,377)(331,379)(332,381)(333,380)(334,382)(335,384)
(336,383);
s1 := Sym(384)!( 1,147)( 2,146)( 3,145)( 4,150)( 5,149)( 6,148)( 7,153)
( 8,152)( 9,151)( 10,156)( 11,155)( 12,154)( 13,159)( 14,158)( 15,157)
( 16,162)( 17,161)( 18,160)( 19,165)( 20,164)( 21,163)( 22,168)( 23,167)
( 24,166)( 25,171)( 26,170)( 27,169)( 28,174)( 29,173)( 30,172)( 31,177)
( 32,176)( 33,175)( 34,180)( 35,179)( 36,178)( 37,183)( 38,182)( 39,181)
( 40,186)( 41,185)( 42,184)( 43,189)( 44,188)( 45,187)( 46,192)( 47,191)
( 48,190)( 49, 99)( 50, 98)( 51, 97)( 52,102)( 53,101)( 54,100)( 55,105)
( 56,104)( 57,103)( 58,108)( 59,107)( 60,106)( 61,111)( 62,110)( 63,109)
( 64,114)( 65,113)( 66,112)( 67,117)( 68,116)( 69,115)( 70,120)( 71,119)
( 72,118)( 73,123)( 74,122)( 75,121)( 76,126)( 77,125)( 78,124)( 79,129)
( 80,128)( 81,127)( 82,132)( 83,131)( 84,130)( 85,135)( 86,134)( 87,133)
( 88,138)( 89,137)( 90,136)( 91,141)( 92,140)( 93,139)( 94,144)( 95,143)
( 96,142)(193,339)(194,338)(195,337)(196,342)(197,341)(198,340)(199,345)
(200,344)(201,343)(202,348)(203,347)(204,346)(205,351)(206,350)(207,349)
(208,354)(209,353)(210,352)(211,357)(212,356)(213,355)(214,360)(215,359)
(216,358)(217,363)(218,362)(219,361)(220,366)(221,365)(222,364)(223,369)
(224,368)(225,367)(226,372)(227,371)(228,370)(229,375)(230,374)(231,373)
(232,378)(233,377)(234,376)(235,381)(236,380)(237,379)(238,384)(239,383)
(240,382)(241,291)(242,290)(243,289)(244,294)(245,293)(246,292)(247,297)
(248,296)(249,295)(250,300)(251,299)(252,298)(253,303)(254,302)(255,301)
(256,306)(257,305)(258,304)(259,309)(260,308)(261,307)(262,312)(263,311)
(264,310)(265,315)(266,314)(267,313)(268,318)(269,317)(270,316)(271,321)
(272,320)(273,319)(274,324)(275,323)(276,322)(277,327)(278,326)(279,325)
(280,330)(281,329)(282,328)(283,333)(284,332)(285,331)(286,336)(287,335)
(288,334);
s2 := Sym(384)!( 7, 10)( 8, 11)( 9, 12)( 19, 22)( 20, 23)( 21, 24)( 25, 31)
( 26, 32)( 27, 33)( 28, 34)( 29, 35)( 30, 36)( 37, 43)( 38, 44)( 39, 45)
( 40, 46)( 41, 47)( 42, 48)( 55, 58)( 56, 59)( 57, 60)( 67, 70)( 68, 71)
( 69, 72)( 73, 79)( 74, 80)( 75, 81)( 76, 82)( 77, 83)( 78, 84)( 85, 91)
( 86, 92)( 87, 93)( 88, 94)( 89, 95)( 90, 96)( 97,109)( 98,110)( 99,111)
(100,112)(101,113)(102,114)(103,118)(104,119)(105,120)(106,115)(107,116)
(108,117)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,133)
(128,134)(129,135)(130,136)(131,137)(132,138)(145,157)(146,158)(147,159)
(148,160)(149,161)(150,162)(151,166)(152,167)(153,168)(154,163)(155,164)
(156,165)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)
(176,182)(177,183)(178,184)(179,185)(180,186)(193,217)(194,218)(195,219)
(196,220)(197,221)(198,222)(199,226)(200,227)(201,228)(202,223)(203,224)
(204,225)(205,229)(206,230)(207,231)(208,232)(209,233)(210,234)(211,238)
(212,239)(213,240)(214,235)(215,236)(216,237)(241,265)(242,266)(243,267)
(244,268)(245,269)(246,270)(247,274)(248,275)(249,276)(250,271)(251,272)
(252,273)(253,277)(254,278)(255,279)(256,280)(257,281)(258,282)(259,286)
(260,287)(261,288)(262,283)(263,284)(264,285)(289,325)(290,326)(291,327)
(292,328)(293,329)(294,330)(295,334)(296,335)(297,336)(298,331)(299,332)
(300,333)(301,313)(302,314)(303,315)(304,316)(305,317)(306,318)(307,322)
(308,323)(309,324)(310,319)(311,320)(312,321)(337,373)(338,374)(339,375)
(340,376)(341,377)(342,378)(343,382)(344,383)(345,384)(346,379)(347,380)
(348,381)(349,361)(350,362)(351,363)(352,364)(353,365)(354,366)(355,370)
(356,371)(357,372)(358,367)(359,368)(360,369);
s3 := Sym(384)!( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,202)
( 8,203)( 9,204)( 10,199)( 11,200)( 12,201)( 13,205)( 14,206)( 15,207)
( 16,208)( 17,209)( 18,210)( 19,214)( 20,215)( 21,216)( 22,211)( 23,212)
( 24,213)( 25,223)( 26,224)( 27,225)( 28,226)( 29,227)( 30,228)( 31,217)
( 32,218)( 33,219)( 34,220)( 35,221)( 36,222)( 37,235)( 38,236)( 39,237)
( 40,238)( 41,239)( 42,240)( 43,229)( 44,230)( 45,231)( 46,232)( 47,233)
( 48,234)( 49,241)( 50,242)( 51,243)( 52,244)( 53,245)( 54,246)( 55,250)
( 56,251)( 57,252)( 58,247)( 59,248)( 60,249)( 61,253)( 62,254)( 63,255)
( 64,256)( 65,257)( 66,258)( 67,262)( 68,263)( 69,264)( 70,259)( 71,260)
( 72,261)( 73,271)( 74,272)( 75,273)( 76,274)( 77,275)( 78,276)( 79,265)
( 80,266)( 81,267)( 82,268)( 83,269)( 84,270)( 85,283)( 86,284)( 87,285)
( 88,286)( 89,287)( 90,288)( 91,277)( 92,278)( 93,279)( 94,280)( 95,281)
( 96,282)( 97,289)( 98,290)( 99,291)(100,292)(101,293)(102,294)(103,298)
(104,299)(105,300)(106,295)(107,296)(108,297)(109,301)(110,302)(111,303)
(112,304)(113,305)(114,306)(115,310)(116,311)(117,312)(118,307)(119,308)
(120,309)(121,319)(122,320)(123,321)(124,322)(125,323)(126,324)(127,313)
(128,314)(129,315)(130,316)(131,317)(132,318)(133,331)(134,332)(135,333)
(136,334)(137,335)(138,336)(139,325)(140,326)(141,327)(142,328)(143,329)
(144,330)(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,346)
(152,347)(153,348)(154,343)(155,344)(156,345)(157,349)(158,350)(159,351)
(160,352)(161,353)(162,354)(163,358)(164,359)(165,360)(166,355)(167,356)
(168,357)(169,367)(170,368)(171,369)(172,370)(173,371)(174,372)(175,361)
(176,362)(177,363)(178,364)(179,365)(180,366)(181,379)(182,380)(183,381)
(184,382)(185,383)(186,384)(187,373)(188,374)(189,375)(190,376)(191,377)
(192,378);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope