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