Polytope of Type {3,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12,4}*768d
if this polytope has a name.
Group : SmallGroup(768,1088556)
Rank : 4
Schlafli Type : {3,12,4}
Number of vertices, edges, etc : 4, 48, 64, 8
Order of s0s1s2s3 : 4
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}*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 := ( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,103)( 6,104)( 7,101)( 8,102)
( 9,108)( 10,107)( 11,106)( 12,105)( 13,110)( 14,109)( 15,112)( 16,111)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,122)( 22,121)( 23,124)( 24,123)
( 25,118)( 26,117)( 27,120)( 28,119)( 29,116)( 30,115)( 31,114)( 32,113)
( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)( 40,166)
( 41,172)( 42,171)( 43,170)( 44,169)( 45,174)( 46,173)( 47,176)( 48,175)
( 49,192)( 50,191)( 51,190)( 52,189)( 53,186)( 54,185)( 55,188)( 56,187)
( 57,182)( 58,181)( 59,184)( 60,183)( 61,180)( 62,179)( 63,178)( 64,177)
( 65,129)( 66,130)( 67,131)( 68,132)( 69,135)( 70,136)( 71,133)( 72,134)
( 73,140)( 74,139)( 75,138)( 76,137)( 77,142)( 78,141)( 79,144)( 80,143)
( 81,160)( 82,159)( 83,158)( 84,157)( 85,154)( 86,153)( 87,156)( 88,155)
( 89,150)( 90,149)( 91,152)( 92,151)( 93,148)( 94,147)( 95,146)( 96,145)
(193,290)(194,289)(195,292)(196,291)(197,296)(198,295)(199,294)(200,293)
(201,299)(202,300)(203,297)(204,298)(205,301)(206,302)(207,303)(208,304)
(209,319)(210,320)(211,317)(212,318)(213,313)(214,314)(215,315)(216,316)
(217,309)(218,310)(219,311)(220,312)(221,307)(222,308)(223,305)(224,306)
(225,354)(226,353)(227,356)(228,355)(229,360)(230,359)(231,358)(232,357)
(233,363)(234,364)(235,361)(236,362)(237,365)(238,366)(239,367)(240,368)
(241,383)(242,384)(243,381)(244,382)(245,377)(246,378)(247,379)(248,380)
(249,373)(250,374)(251,375)(252,376)(253,371)(254,372)(255,369)(256,370)
(257,322)(258,321)(259,324)(260,323)(261,328)(262,327)(263,326)(264,325)
(265,331)(266,332)(267,329)(268,330)(269,333)(270,334)(271,335)(272,336)
(273,351)(274,352)(275,349)(276,350)(277,345)(278,346)(279,347)(280,348)
(281,341)(282,342)(283,343)(284,344)(285,339)(286,340)(287,337)(288,338);;
s1 := ( 1,161)( 2,162)( 3,165)( 4,166)( 5,163)( 6,164)( 7,167)( 8,168)
( 9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)
( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)
( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)
( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)
( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)
( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)
( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)
( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)
( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)
( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)
( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)
(193,354)(194,353)(195,358)(196,357)(197,356)(198,355)(199,360)(200,359)
(201,373)(202,374)(203,369)(204,370)(205,375)(206,376)(207,371)(208,372)
(209,363)(210,364)(211,367)(212,368)(213,361)(214,362)(215,365)(216,366)
(217,383)(218,384)(219,379)(220,380)(221,381)(222,382)(223,377)(224,378)
(225,322)(226,321)(227,326)(228,325)(229,324)(230,323)(231,328)(232,327)
(233,341)(234,342)(235,337)(236,338)(237,343)(238,344)(239,339)(240,340)
(241,331)(242,332)(243,335)(244,336)(245,329)(246,330)(247,333)(248,334)
(249,351)(250,352)(251,347)(252,348)(253,349)(254,350)(255,345)(256,346)
(257,290)(258,289)(259,294)(260,293)(261,292)(262,291)(263,296)(264,295)
(265,309)(266,310)(267,305)(268,306)(269,311)(270,312)(271,307)(272,308)
(273,299)(274,300)(275,303)(276,304)(277,297)(278,298)(279,301)(280,302)
(281,319)(282,320)(283,315)(284,316)(285,317)(286,318)(287,313)(288,314);;
s2 := ( 1,304)( 2,303)( 3,302)( 4,301)( 5,297)( 6,298)( 7,299)( 8,300)
( 9,293)( 10,294)( 11,295)( 12,296)( 13,292)( 14,291)( 15,290)( 16,289)
( 17,306)( 18,305)( 19,308)( 20,307)( 21,311)( 22,312)( 23,309)( 24,310)
( 25,316)( 26,315)( 27,314)( 28,313)( 29,317)( 30,318)( 31,319)( 32,320)
( 33,368)( 34,367)( 35,366)( 36,365)( 37,361)( 38,362)( 39,363)( 40,364)
( 41,357)( 42,358)( 43,359)( 44,360)( 45,356)( 46,355)( 47,354)( 48,353)
( 49,370)( 50,369)( 51,372)( 52,371)( 53,375)( 54,376)( 55,373)( 56,374)
( 57,380)( 58,379)( 59,378)( 60,377)( 61,381)( 62,382)( 63,383)( 64,384)
( 65,336)( 66,335)( 67,334)( 68,333)( 69,329)( 70,330)( 71,331)( 72,332)
( 73,325)( 74,326)( 75,327)( 76,328)( 77,324)( 78,323)( 79,322)( 80,321)
( 81,338)( 82,337)( 83,340)( 84,339)( 85,343)( 86,344)( 87,341)( 88,342)
( 89,348)( 90,347)( 91,346)( 92,345)( 93,349)( 94,350)( 95,351)( 96,352)
( 97,208)( 98,207)( 99,206)(100,205)(101,201)(102,202)(103,203)(104,204)
(105,197)(106,198)(107,199)(108,200)(109,196)(110,195)(111,194)(112,193)
(113,210)(114,209)(115,212)(116,211)(117,215)(118,216)(119,213)(120,214)
(121,220)(122,219)(123,218)(124,217)(125,221)(126,222)(127,223)(128,224)
(129,272)(130,271)(131,270)(132,269)(133,265)(134,266)(135,267)(136,268)
(137,261)(138,262)(139,263)(140,264)(141,260)(142,259)(143,258)(144,257)
(145,274)(146,273)(147,276)(148,275)(149,279)(150,280)(151,277)(152,278)
(153,284)(154,283)(155,282)(156,281)(157,285)(158,286)(159,287)(160,288)
(161,240)(162,239)(163,238)(164,237)(165,233)(166,234)(167,235)(168,236)
(169,229)(170,230)(171,231)(172,232)(173,228)(174,227)(175,226)(176,225)
(177,242)(178,241)(179,244)(180,243)(181,247)(182,248)(183,245)(184,246)
(185,252)(186,251)(187,250)(188,249)(189,253)(190,254)(191,255)(192,256);;
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, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s3*s0*s2*s1*s3*s0*s2*s1*s3*s2*s0*s1*s3*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1, 97)( 2, 98)( 3, 99)( 4,100)( 5,103)( 6,104)( 7,101)
( 8,102)( 9,108)( 10,107)( 11,106)( 12,105)( 13,110)( 14,109)( 15,112)
( 16,111)( 17,128)( 18,127)( 19,126)( 20,125)( 21,122)( 22,121)( 23,124)
( 24,123)( 25,118)( 26,117)( 27,120)( 28,119)( 29,116)( 30,115)( 31,114)
( 32,113)( 33,161)( 34,162)( 35,163)( 36,164)( 37,167)( 38,168)( 39,165)
( 40,166)( 41,172)( 42,171)( 43,170)( 44,169)( 45,174)( 46,173)( 47,176)
( 48,175)( 49,192)( 50,191)( 51,190)( 52,189)( 53,186)( 54,185)( 55,188)
( 56,187)( 57,182)( 58,181)( 59,184)( 60,183)( 61,180)( 62,179)( 63,178)
( 64,177)( 65,129)( 66,130)( 67,131)( 68,132)( 69,135)( 70,136)( 71,133)
( 72,134)( 73,140)( 74,139)( 75,138)( 76,137)( 77,142)( 78,141)( 79,144)
( 80,143)( 81,160)( 82,159)( 83,158)( 84,157)( 85,154)( 86,153)( 87,156)
( 88,155)( 89,150)( 90,149)( 91,152)( 92,151)( 93,148)( 94,147)( 95,146)
( 96,145)(193,290)(194,289)(195,292)(196,291)(197,296)(198,295)(199,294)
(200,293)(201,299)(202,300)(203,297)(204,298)(205,301)(206,302)(207,303)
(208,304)(209,319)(210,320)(211,317)(212,318)(213,313)(214,314)(215,315)
(216,316)(217,309)(218,310)(219,311)(220,312)(221,307)(222,308)(223,305)
(224,306)(225,354)(226,353)(227,356)(228,355)(229,360)(230,359)(231,358)
(232,357)(233,363)(234,364)(235,361)(236,362)(237,365)(238,366)(239,367)
(240,368)(241,383)(242,384)(243,381)(244,382)(245,377)(246,378)(247,379)
(248,380)(249,373)(250,374)(251,375)(252,376)(253,371)(254,372)(255,369)
(256,370)(257,322)(258,321)(259,324)(260,323)(261,328)(262,327)(263,326)
(264,325)(265,331)(266,332)(267,329)(268,330)(269,333)(270,334)(271,335)
(272,336)(273,351)(274,352)(275,349)(276,350)(277,345)(278,346)(279,347)
(280,348)(281,341)(282,342)(283,343)(284,344)(285,339)(286,340)(287,337)
(288,338);
s1 := Sym(384)!( 1,161)( 2,162)( 3,165)( 4,166)( 5,163)( 6,164)( 7,167)
( 8,168)( 9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)
( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)
( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)
( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)
( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)
( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)
( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)
( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)
( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)
( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)
( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)
( 96,121)(193,354)(194,353)(195,358)(196,357)(197,356)(198,355)(199,360)
(200,359)(201,373)(202,374)(203,369)(204,370)(205,375)(206,376)(207,371)
(208,372)(209,363)(210,364)(211,367)(212,368)(213,361)(214,362)(215,365)
(216,366)(217,383)(218,384)(219,379)(220,380)(221,381)(222,382)(223,377)
(224,378)(225,322)(226,321)(227,326)(228,325)(229,324)(230,323)(231,328)
(232,327)(233,341)(234,342)(235,337)(236,338)(237,343)(238,344)(239,339)
(240,340)(241,331)(242,332)(243,335)(244,336)(245,329)(246,330)(247,333)
(248,334)(249,351)(250,352)(251,347)(252,348)(253,349)(254,350)(255,345)
(256,346)(257,290)(258,289)(259,294)(260,293)(261,292)(262,291)(263,296)
(264,295)(265,309)(266,310)(267,305)(268,306)(269,311)(270,312)(271,307)
(272,308)(273,299)(274,300)(275,303)(276,304)(277,297)(278,298)(279,301)
(280,302)(281,319)(282,320)(283,315)(284,316)(285,317)(286,318)(287,313)
(288,314);
s2 := Sym(384)!( 1,304)( 2,303)( 3,302)( 4,301)( 5,297)( 6,298)( 7,299)
( 8,300)( 9,293)( 10,294)( 11,295)( 12,296)( 13,292)( 14,291)( 15,290)
( 16,289)( 17,306)( 18,305)( 19,308)( 20,307)( 21,311)( 22,312)( 23,309)
( 24,310)( 25,316)( 26,315)( 27,314)( 28,313)( 29,317)( 30,318)( 31,319)
( 32,320)( 33,368)( 34,367)( 35,366)( 36,365)( 37,361)( 38,362)( 39,363)
( 40,364)( 41,357)( 42,358)( 43,359)( 44,360)( 45,356)( 46,355)( 47,354)
( 48,353)( 49,370)( 50,369)( 51,372)( 52,371)( 53,375)( 54,376)( 55,373)
( 56,374)( 57,380)( 58,379)( 59,378)( 60,377)( 61,381)( 62,382)( 63,383)
( 64,384)( 65,336)( 66,335)( 67,334)( 68,333)( 69,329)( 70,330)( 71,331)
( 72,332)( 73,325)( 74,326)( 75,327)( 76,328)( 77,324)( 78,323)( 79,322)
( 80,321)( 81,338)( 82,337)( 83,340)( 84,339)( 85,343)( 86,344)( 87,341)
( 88,342)( 89,348)( 90,347)( 91,346)( 92,345)( 93,349)( 94,350)( 95,351)
( 96,352)( 97,208)( 98,207)( 99,206)(100,205)(101,201)(102,202)(103,203)
(104,204)(105,197)(106,198)(107,199)(108,200)(109,196)(110,195)(111,194)
(112,193)(113,210)(114,209)(115,212)(116,211)(117,215)(118,216)(119,213)
(120,214)(121,220)(122,219)(123,218)(124,217)(125,221)(126,222)(127,223)
(128,224)(129,272)(130,271)(131,270)(132,269)(133,265)(134,266)(135,267)
(136,268)(137,261)(138,262)(139,263)(140,264)(141,260)(142,259)(143,258)
(144,257)(145,274)(146,273)(147,276)(148,275)(149,279)(150,280)(151,277)
(152,278)(153,284)(154,283)(155,282)(156,281)(157,285)(158,286)(159,287)
(160,288)(161,240)(162,239)(163,238)(164,237)(165,233)(166,234)(167,235)
(168,236)(169,229)(170,230)(171,231)(172,232)(173,228)(174,227)(175,226)
(176,225)(177,242)(178,241)(179,244)(180,243)(181,247)(182,248)(183,245)
(184,246)(185,252)(186,251)(187,250)(188,249)(189,253)(190,254)(191,255)
(192,256);
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1,
s3*s0*s2*s1*s3*s0*s2*s1*s3*s2*s0*s1*s3*s2*s0*s1 >;
References : None.
to this polytope