Polytope of Type {4,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,4}*768m
if this polytope has a name.
Group : SmallGroup(768,1090188)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 4, 48, 48, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-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 : {4,12,4}*384f, {4,12,4}*384g, {4,6,4}*384c
4-fold quotients : {4,12,2}*192b, {4,3,4}*192a, {4,6,4}*192d, {4,6,4}*192e
8-fold quotients : {4,6,2}*96c, {4,3,4}*96
16-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)( 14, 15)
( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)( 30, 31)
( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)( 62, 63)
( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)( 78, 79)
( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)( 94, 95)
( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)(110,111)
(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)(126,127)
(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)(142,143)
(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)(158,159)
(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)(174,175)
(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)(190,191)
(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)(206,207)
(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)(222,223)
(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)(238,239)
(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)(254,255)
(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)(270,271)
(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)(286,287)
(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)(302,303)
(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)(318,319)
(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)(334,335)
(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)(350,351)
(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)(366,367)
(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)(382,383);;
s1 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)
(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)(108,159)
(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)
(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)(124,191)
(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)
(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)(140,175)
(141,169)(142,170)(143,172)(144,171)(195,196)(199,200)(201,205)(202,206)
(203,208)(204,207)(209,225)(210,226)(211,228)(212,227)(213,229)(214,230)
(215,232)(216,231)(217,237)(218,238)(219,240)(220,239)(221,233)(222,234)
(223,236)(224,235)(243,244)(247,248)(249,253)(250,254)(251,256)(252,255)
(257,273)(258,274)(259,276)(260,275)(261,277)(262,278)(263,280)(264,279)
(265,285)(266,286)(267,288)(268,287)(269,281)(270,282)(271,284)(272,283)
(289,337)(290,338)(291,340)(292,339)(293,341)(294,342)(295,344)(296,343)
(297,349)(298,350)(299,352)(300,351)(301,345)(302,346)(303,348)(304,347)
(305,369)(306,370)(307,372)(308,371)(309,373)(310,374)(311,376)(312,375)
(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)(320,379)
(321,353)(322,354)(323,356)(324,355)(325,357)(326,358)(327,360)(328,359)
(329,365)(330,366)(331,368)(332,367)(333,361)(334,362)(335,364)(336,363);;
s2 := ( 1,177)( 2,179)( 3,178)( 4,180)( 5,185)( 6,187)( 7,186)( 8,188)
( 9,181)( 10,183)( 11,182)( 12,184)( 13,189)( 14,191)( 15,190)( 16,192)
( 17,161)( 18,163)( 19,162)( 20,164)( 21,169)( 22,171)( 23,170)( 24,172)
( 25,165)( 26,167)( 27,166)( 28,168)( 29,173)( 30,175)( 31,174)( 32,176)
( 33,145)( 34,147)( 35,146)( 36,148)( 37,153)( 38,155)( 39,154)( 40,156)
( 41,149)( 42,151)( 43,150)( 44,152)( 45,157)( 46,159)( 47,158)( 48,160)
( 49,129)( 50,131)( 51,130)( 52,132)( 53,137)( 54,139)( 55,138)( 56,140)
( 57,133)( 58,135)( 59,134)( 60,136)( 61,141)( 62,143)( 63,142)( 64,144)
( 65,113)( 66,115)( 67,114)( 68,116)( 69,121)( 70,123)( 71,122)( 72,124)
( 73,117)( 74,119)( 75,118)( 76,120)( 77,125)( 78,127)( 79,126)( 80,128)
( 81, 97)( 82, 99)( 83, 98)( 84,100)( 85,105)( 86,107)( 87,106)( 88,108)
( 89,101)( 90,103)( 91,102)( 92,104)( 93,109)( 94,111)( 95,110)( 96,112)
(193,369)(194,371)(195,370)(196,372)(197,377)(198,379)(199,378)(200,380)
(201,373)(202,375)(203,374)(204,376)(205,381)(206,383)(207,382)(208,384)
(209,353)(210,355)(211,354)(212,356)(213,361)(214,363)(215,362)(216,364)
(217,357)(218,359)(219,358)(220,360)(221,365)(222,367)(223,366)(224,368)
(225,337)(226,339)(227,338)(228,340)(229,345)(230,347)(231,346)(232,348)
(233,341)(234,343)(235,342)(236,344)(237,349)(238,351)(239,350)(240,352)
(241,321)(242,323)(243,322)(244,324)(245,329)(246,331)(247,330)(248,332)
(249,325)(250,327)(251,326)(252,328)(253,333)(254,335)(255,334)(256,336)
(257,305)(258,307)(259,306)(260,308)(261,313)(262,315)(263,314)(264,316)
(265,309)(266,311)(267,310)(268,312)(269,317)(270,319)(271,318)(272,320)
(273,289)(274,291)(275,290)(276,292)(277,297)(278,299)(279,298)(280,300)
(281,293)(282,295)(283,294)(284,296)(285,301)(286,303)(287,302)(288,304);;
s3 := ( 1,197)( 2,198)( 3,199)( 4,200)( 5,193)( 6,194)( 7,195)( 8,196)
( 9,205)( 10,206)( 11,207)( 12,208)( 13,201)( 14,202)( 15,203)( 16,204)
( 17,213)( 18,214)( 19,215)( 20,216)( 21,209)( 22,210)( 23,211)( 24,212)
( 25,221)( 26,222)( 27,223)( 28,224)( 29,217)( 30,218)( 31,219)( 32,220)
( 33,229)( 34,230)( 35,231)( 36,232)( 37,225)( 38,226)( 39,227)( 40,228)
( 41,237)( 42,238)( 43,239)( 44,240)( 45,233)( 46,234)( 47,235)( 48,236)
( 49,245)( 50,246)( 51,247)( 52,248)( 53,241)( 54,242)( 55,243)( 56,244)
( 57,253)( 58,254)( 59,255)( 60,256)( 61,249)( 62,250)( 63,251)( 64,252)
( 65,261)( 66,262)( 67,263)( 68,264)( 69,257)( 70,258)( 71,259)( 72,260)
( 73,269)( 74,270)( 75,271)( 76,272)( 77,265)( 78,266)( 79,267)( 80,268)
( 81,277)( 82,278)( 83,279)( 84,280)( 85,273)( 86,274)( 87,275)( 88,276)
( 89,285)( 90,286)( 91,287)( 92,288)( 93,281)( 94,282)( 95,283)( 96,284)
( 97,293)( 98,294)( 99,295)(100,296)(101,289)(102,290)(103,291)(104,292)
(105,301)(106,302)(107,303)(108,304)(109,297)(110,298)(111,299)(112,300)
(113,309)(114,310)(115,311)(116,312)(117,305)(118,306)(119,307)(120,308)
(121,317)(122,318)(123,319)(124,320)(125,313)(126,314)(127,315)(128,316)
(129,325)(130,326)(131,327)(132,328)(133,321)(134,322)(135,323)(136,324)
(137,333)(138,334)(139,335)(140,336)(141,329)(142,330)(143,331)(144,332)
(145,341)(146,342)(147,343)(148,344)(149,337)(150,338)(151,339)(152,340)
(153,349)(154,350)(155,351)(156,352)(157,345)(158,346)(159,347)(160,348)
(161,357)(162,358)(163,359)(164,360)(165,353)(166,354)(167,355)(168,356)
(169,365)(170,366)(171,367)(172,368)(173,361)(174,362)(175,363)(176,364)
(177,373)(178,374)(179,375)(180,376)(181,369)(182,370)(183,371)(184,372)
(185,381)(186,382)(187,383)(188,384)(189,377)(190,378)(191,379)(192,380);;
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*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9, 12)( 10, 11)( 13, 16)
( 14, 15)( 17, 20)( 18, 19)( 21, 24)( 22, 23)( 25, 28)( 26, 27)( 29, 32)
( 30, 31)( 33, 36)( 34, 35)( 37, 40)( 38, 39)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 52)( 50, 51)( 53, 56)( 54, 55)( 57, 60)( 58, 59)( 61, 64)
( 62, 63)( 65, 68)( 66, 67)( 69, 72)( 70, 71)( 73, 76)( 74, 75)( 77, 80)
( 78, 79)( 81, 84)( 82, 83)( 85, 88)( 86, 87)( 89, 92)( 90, 91)( 93, 96)
( 94, 95)( 97,100)( 98, 99)(101,104)(102,103)(105,108)(106,107)(109,112)
(110,111)(113,116)(114,115)(117,120)(118,119)(121,124)(122,123)(125,128)
(126,127)(129,132)(130,131)(133,136)(134,135)(137,140)(138,139)(141,144)
(142,143)(145,148)(146,147)(149,152)(150,151)(153,156)(154,155)(157,160)
(158,159)(161,164)(162,163)(165,168)(166,167)(169,172)(170,171)(173,176)
(174,175)(177,180)(178,179)(181,184)(182,183)(185,188)(186,187)(189,192)
(190,191)(193,196)(194,195)(197,200)(198,199)(201,204)(202,203)(205,208)
(206,207)(209,212)(210,211)(213,216)(214,215)(217,220)(218,219)(221,224)
(222,223)(225,228)(226,227)(229,232)(230,231)(233,236)(234,235)(237,240)
(238,239)(241,244)(242,243)(245,248)(246,247)(249,252)(250,251)(253,256)
(254,255)(257,260)(258,259)(261,264)(262,263)(265,268)(266,267)(269,272)
(270,271)(273,276)(274,275)(277,280)(278,279)(281,284)(282,283)(285,288)
(286,287)(289,292)(290,291)(293,296)(294,295)(297,300)(298,299)(301,304)
(302,303)(305,308)(306,307)(309,312)(310,311)(313,316)(314,315)(317,320)
(318,319)(321,324)(322,323)(325,328)(326,327)(329,332)(330,331)(333,336)
(334,335)(337,340)(338,339)(341,344)(342,343)(345,348)(346,347)(349,352)
(350,351)(353,356)(354,355)(357,360)(358,359)(361,364)(362,363)(365,368)
(366,367)(369,372)(370,371)(373,376)(374,375)(377,380)(378,379)(381,384)
(382,383);
s1 := Sym(384)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)
(100,147)(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)
(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)
(116,179)(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)
(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)
(132,163)(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)
(140,175)(141,169)(142,170)(143,172)(144,171)(195,196)(199,200)(201,205)
(202,206)(203,208)(204,207)(209,225)(210,226)(211,228)(212,227)(213,229)
(214,230)(215,232)(216,231)(217,237)(218,238)(219,240)(220,239)(221,233)
(222,234)(223,236)(224,235)(243,244)(247,248)(249,253)(250,254)(251,256)
(252,255)(257,273)(258,274)(259,276)(260,275)(261,277)(262,278)(263,280)
(264,279)(265,285)(266,286)(267,288)(268,287)(269,281)(270,282)(271,284)
(272,283)(289,337)(290,338)(291,340)(292,339)(293,341)(294,342)(295,344)
(296,343)(297,349)(298,350)(299,352)(300,351)(301,345)(302,346)(303,348)
(304,347)(305,369)(306,370)(307,372)(308,371)(309,373)(310,374)(311,376)
(312,375)(313,381)(314,382)(315,384)(316,383)(317,377)(318,378)(319,380)
(320,379)(321,353)(322,354)(323,356)(324,355)(325,357)(326,358)(327,360)
(328,359)(329,365)(330,366)(331,368)(332,367)(333,361)(334,362)(335,364)
(336,363);
s2 := Sym(384)!( 1,177)( 2,179)( 3,178)( 4,180)( 5,185)( 6,187)( 7,186)
( 8,188)( 9,181)( 10,183)( 11,182)( 12,184)( 13,189)( 14,191)( 15,190)
( 16,192)( 17,161)( 18,163)( 19,162)( 20,164)( 21,169)( 22,171)( 23,170)
( 24,172)( 25,165)( 26,167)( 27,166)( 28,168)( 29,173)( 30,175)( 31,174)
( 32,176)( 33,145)( 34,147)( 35,146)( 36,148)( 37,153)( 38,155)( 39,154)
( 40,156)( 41,149)( 42,151)( 43,150)( 44,152)( 45,157)( 46,159)( 47,158)
( 48,160)( 49,129)( 50,131)( 51,130)( 52,132)( 53,137)( 54,139)( 55,138)
( 56,140)( 57,133)( 58,135)( 59,134)( 60,136)( 61,141)( 62,143)( 63,142)
( 64,144)( 65,113)( 66,115)( 67,114)( 68,116)( 69,121)( 70,123)( 71,122)
( 72,124)( 73,117)( 74,119)( 75,118)( 76,120)( 77,125)( 78,127)( 79,126)
( 80,128)( 81, 97)( 82, 99)( 83, 98)( 84,100)( 85,105)( 86,107)( 87,106)
( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)( 93,109)( 94,111)( 95,110)
( 96,112)(193,369)(194,371)(195,370)(196,372)(197,377)(198,379)(199,378)
(200,380)(201,373)(202,375)(203,374)(204,376)(205,381)(206,383)(207,382)
(208,384)(209,353)(210,355)(211,354)(212,356)(213,361)(214,363)(215,362)
(216,364)(217,357)(218,359)(219,358)(220,360)(221,365)(222,367)(223,366)
(224,368)(225,337)(226,339)(227,338)(228,340)(229,345)(230,347)(231,346)
(232,348)(233,341)(234,343)(235,342)(236,344)(237,349)(238,351)(239,350)
(240,352)(241,321)(242,323)(243,322)(244,324)(245,329)(246,331)(247,330)
(248,332)(249,325)(250,327)(251,326)(252,328)(253,333)(254,335)(255,334)
(256,336)(257,305)(258,307)(259,306)(260,308)(261,313)(262,315)(263,314)
(264,316)(265,309)(266,311)(267,310)(268,312)(269,317)(270,319)(271,318)
(272,320)(273,289)(274,291)(275,290)(276,292)(277,297)(278,299)(279,298)
(280,300)(281,293)(282,295)(283,294)(284,296)(285,301)(286,303)(287,302)
(288,304);
s3 := Sym(384)!( 1,197)( 2,198)( 3,199)( 4,200)( 5,193)( 6,194)( 7,195)
( 8,196)( 9,205)( 10,206)( 11,207)( 12,208)( 13,201)( 14,202)( 15,203)
( 16,204)( 17,213)( 18,214)( 19,215)( 20,216)( 21,209)( 22,210)( 23,211)
( 24,212)( 25,221)( 26,222)( 27,223)( 28,224)( 29,217)( 30,218)( 31,219)
( 32,220)( 33,229)( 34,230)( 35,231)( 36,232)( 37,225)( 38,226)( 39,227)
( 40,228)( 41,237)( 42,238)( 43,239)( 44,240)( 45,233)( 46,234)( 47,235)
( 48,236)( 49,245)( 50,246)( 51,247)( 52,248)( 53,241)( 54,242)( 55,243)
( 56,244)( 57,253)( 58,254)( 59,255)( 60,256)( 61,249)( 62,250)( 63,251)
( 64,252)( 65,261)( 66,262)( 67,263)( 68,264)( 69,257)( 70,258)( 71,259)
( 72,260)( 73,269)( 74,270)( 75,271)( 76,272)( 77,265)( 78,266)( 79,267)
( 80,268)( 81,277)( 82,278)( 83,279)( 84,280)( 85,273)( 86,274)( 87,275)
( 88,276)( 89,285)( 90,286)( 91,287)( 92,288)( 93,281)( 94,282)( 95,283)
( 96,284)( 97,293)( 98,294)( 99,295)(100,296)(101,289)(102,290)(103,291)
(104,292)(105,301)(106,302)(107,303)(108,304)(109,297)(110,298)(111,299)
(112,300)(113,309)(114,310)(115,311)(116,312)(117,305)(118,306)(119,307)
(120,308)(121,317)(122,318)(123,319)(124,320)(125,313)(126,314)(127,315)
(128,316)(129,325)(130,326)(131,327)(132,328)(133,321)(134,322)(135,323)
(136,324)(137,333)(138,334)(139,335)(140,336)(141,329)(142,330)(143,331)
(144,332)(145,341)(146,342)(147,343)(148,344)(149,337)(150,338)(151,339)
(152,340)(153,349)(154,350)(155,351)(156,352)(157,345)(158,346)(159,347)
(160,348)(161,357)(162,358)(163,359)(164,360)(165,353)(166,354)(167,355)
(168,356)(169,365)(170,366)(171,367)(172,368)(173,361)(174,362)(175,363)
(176,364)(177,373)(178,374)(179,375)(180,376)(181,369)(182,370)(183,371)
(184,372)(185,381)(186,382)(187,383)(188,384)(189,377)(190,378)(191,379)
(192,380);
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*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s0*s1, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope