Polytope of Type {4,4,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,12}*768e
if this polytope has a name.
Group : SmallGroup(768,1088766)
Rank : 4
Schlafli Type : {4,4,12}
Number of vertices, edges, etc : 4, 16, 48, 24
Order of s0s1s2s3 : 12
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 : {2,4,12}*384b, {4,4,6}*384d
4-fold quotients : {4,2,12}*192, {2,4,12}*192b, {2,4,12}*192c, {4,4,3}*192b, {2,4,6}*192
8-fold quotients : {2,2,12}*96, {4,2,6}*96, {2,4,3}*96, {2,4,6}*96b, {2,4,6}*96c
12-fold quotients : {4,2,4}*64
16-fold quotients : {4,2,3}*48, {2,4,3}*48, {2,2,6}*48
24-fold quotients : {2,2,4}*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, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)( 8,104)
( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)( 16,112)
( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)( 24,120)
( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)( 32,128)
( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)( 40,136)
( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)( 48,144)
( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)( 56,176)
( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)( 64,184)
( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)( 72,192)
( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)( 80,152)
( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)( 88,160)
( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)( 96,168)
(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)(200,296)
(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)(208,304)
(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)(216,312)
(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)(224,320)
(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)(232,328)
(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)(240,336)
(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)(248,368)
(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)(256,376)
(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)(264,384)
(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)(272,344)
(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)(280,352)
(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)(288,360);;
s1 := ( 1, 52)( 2, 51)( 3, 50)( 4, 49)( 5, 56)( 6, 55)( 7, 54)( 8, 53)
( 9, 60)( 10, 59)( 11, 58)( 12, 57)( 13, 64)( 14, 63)( 15, 62)( 16, 61)
( 17, 68)( 18, 67)( 19, 66)( 20, 65)( 21, 72)( 22, 71)( 23, 70)( 24, 69)
( 25, 76)( 26, 75)( 27, 74)( 28, 73)( 29, 80)( 30, 79)( 31, 78)( 32, 77)
( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 88)( 38, 87)( 39, 86)( 40, 85)
( 41, 92)( 42, 91)( 43, 90)( 44, 89)( 45, 96)( 46, 95)( 47, 94)( 48, 93)
( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)(104,149)
(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)(112,157)
(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)(120,165)
(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)(128,173)
(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)(136,181)
(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)(144,189)
(193,244)(194,243)(195,242)(196,241)(197,248)(198,247)(199,246)(200,245)
(201,252)(202,251)(203,250)(204,249)(205,256)(206,255)(207,254)(208,253)
(209,260)(210,259)(211,258)(212,257)(213,264)(214,263)(215,262)(216,261)
(217,268)(218,267)(219,266)(220,265)(221,272)(222,271)(223,270)(224,269)
(225,276)(226,275)(227,274)(228,273)(229,280)(230,279)(231,278)(232,277)
(233,284)(234,283)(235,282)(236,281)(237,288)(238,287)(239,286)(240,285)
(289,340)(290,339)(291,338)(292,337)(293,344)(294,343)(295,342)(296,341)
(297,348)(298,347)(299,346)(300,345)(301,352)(302,351)(303,350)(304,349)
(305,356)(306,355)(307,354)(308,353)(309,360)(310,359)(311,358)(312,357)
(313,364)(314,363)(315,362)(316,361)(317,368)(318,367)(319,366)(320,365)
(321,372)(322,371)(323,370)(324,369)(325,376)(326,375)(327,374)(328,373)
(329,380)(330,379)(331,378)(332,377)(333,384)(334,383)(335,382)(336,381);;
s2 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)( 39, 40)
( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)( 55, 60)
( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)( 77, 81)
( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)( 92, 95)
( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)(114,118)
(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)(135,136)
(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)(151,156)
(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)(173,177)
(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)(188,191)
(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)(199,216)(200,215)
(201,209)(202,210)(203,212)(204,211)(217,229)(218,230)(219,232)(220,231)
(221,237)(222,238)(223,240)(224,239)(225,233)(226,234)(227,236)(228,235)
(241,253)(242,254)(243,256)(244,255)(245,261)(246,262)(247,264)(248,263)
(249,257)(250,258)(251,260)(252,259)(265,277)(266,278)(267,280)(268,279)
(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)(276,283)
(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)(296,311)
(297,305)(298,306)(299,308)(300,307)(313,325)(314,326)(315,328)(316,327)
(317,333)(318,334)(319,336)(320,335)(321,329)(322,330)(323,332)(324,331)
(337,349)(338,350)(339,352)(340,351)(341,357)(342,358)(343,360)(344,359)
(345,353)(346,354)(347,356)(348,355)(361,373)(362,374)(363,376)(364,375)
(365,381)(366,382)(367,384)(368,383)(369,377)(370,378)(371,380)(372,379);;
s3 := ( 1,213)( 2,215)( 3,214)( 4,216)( 5,209)( 6,211)( 7,210)( 8,212)
( 9,205)( 10,207)( 11,206)( 12,208)( 13,201)( 14,203)( 15,202)( 16,204)
( 17,197)( 18,199)( 19,198)( 20,200)( 21,193)( 22,195)( 23,194)( 24,196)
( 25,237)( 26,239)( 27,238)( 28,240)( 29,233)( 30,235)( 31,234)( 32,236)
( 33,229)( 34,231)( 35,230)( 36,232)( 37,225)( 38,227)( 39,226)( 40,228)
( 41,221)( 42,223)( 43,222)( 44,224)( 45,217)( 46,219)( 47,218)( 48,220)
( 49,261)( 50,263)( 51,262)( 52,264)( 53,257)( 54,259)( 55,258)( 56,260)
( 57,253)( 58,255)( 59,254)( 60,256)( 61,249)( 62,251)( 63,250)( 64,252)
( 65,245)( 66,247)( 67,246)( 68,248)( 69,241)( 70,243)( 71,242)( 72,244)
( 73,285)( 74,287)( 75,286)( 76,288)( 77,281)( 78,283)( 79,282)( 80,284)
( 81,277)( 82,279)( 83,278)( 84,280)( 85,273)( 86,275)( 87,274)( 88,276)
( 89,269)( 90,271)( 91,270)( 92,272)( 93,265)( 94,267)( 95,266)( 96,268)
( 97,309)( 98,311)( 99,310)(100,312)(101,305)(102,307)(103,306)(104,308)
(105,301)(106,303)(107,302)(108,304)(109,297)(110,299)(111,298)(112,300)
(113,293)(114,295)(115,294)(116,296)(117,289)(118,291)(119,290)(120,292)
(121,333)(122,335)(123,334)(124,336)(125,329)(126,331)(127,330)(128,332)
(129,325)(130,327)(131,326)(132,328)(133,321)(134,323)(135,322)(136,324)
(137,317)(138,319)(139,318)(140,320)(141,313)(142,315)(143,314)(144,316)
(145,357)(146,359)(147,358)(148,360)(149,353)(150,355)(151,354)(152,356)
(153,349)(154,351)(155,350)(156,352)(157,345)(158,347)(159,346)(160,348)
(161,341)(162,343)(163,342)(164,344)(165,337)(166,339)(167,338)(168,340)
(169,381)(170,383)(171,382)(172,384)(173,377)(174,379)(175,378)(176,380)
(177,373)(178,375)(179,374)(180,376)(181,369)(182,371)(183,370)(184,372)
(185,365)(186,367)(187,366)(188,368)(189,361)(190,363)(191,362)(192,364);;
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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, 97)( 2, 98)( 3, 99)( 4,100)( 5,101)( 6,102)( 7,103)
( 8,104)( 9,105)( 10,106)( 11,107)( 12,108)( 13,109)( 14,110)( 15,111)
( 16,112)( 17,113)( 18,114)( 19,115)( 20,116)( 21,117)( 22,118)( 23,119)
( 24,120)( 25,121)( 26,122)( 27,123)( 28,124)( 29,125)( 30,126)( 31,127)
( 32,128)( 33,129)( 34,130)( 35,131)( 36,132)( 37,133)( 38,134)( 39,135)
( 40,136)( 41,137)( 42,138)( 43,139)( 44,140)( 45,141)( 46,142)( 47,143)
( 48,144)( 49,169)( 50,170)( 51,171)( 52,172)( 53,173)( 54,174)( 55,175)
( 56,176)( 57,177)( 58,178)( 59,179)( 60,180)( 61,181)( 62,182)( 63,183)
( 64,184)( 65,185)( 66,186)( 67,187)( 68,188)( 69,189)( 70,190)( 71,191)
( 72,192)( 73,145)( 74,146)( 75,147)( 76,148)( 77,149)( 78,150)( 79,151)
( 80,152)( 81,153)( 82,154)( 83,155)( 84,156)( 85,157)( 86,158)( 87,159)
( 88,160)( 89,161)( 90,162)( 91,163)( 92,164)( 93,165)( 94,166)( 95,167)
( 96,168)(193,289)(194,290)(195,291)(196,292)(197,293)(198,294)(199,295)
(200,296)(201,297)(202,298)(203,299)(204,300)(205,301)(206,302)(207,303)
(208,304)(209,305)(210,306)(211,307)(212,308)(213,309)(214,310)(215,311)
(216,312)(217,313)(218,314)(219,315)(220,316)(221,317)(222,318)(223,319)
(224,320)(225,321)(226,322)(227,323)(228,324)(229,325)(230,326)(231,327)
(232,328)(233,329)(234,330)(235,331)(236,332)(237,333)(238,334)(239,335)
(240,336)(241,361)(242,362)(243,363)(244,364)(245,365)(246,366)(247,367)
(248,368)(249,369)(250,370)(251,371)(252,372)(253,373)(254,374)(255,375)
(256,376)(257,377)(258,378)(259,379)(260,380)(261,381)(262,382)(263,383)
(264,384)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,343)
(272,344)(273,345)(274,346)(275,347)(276,348)(277,349)(278,350)(279,351)
(280,352)(281,353)(282,354)(283,355)(284,356)(285,357)(286,358)(287,359)
(288,360);
s1 := Sym(384)!( 1, 52)( 2, 51)( 3, 50)( 4, 49)( 5, 56)( 6, 55)( 7, 54)
( 8, 53)( 9, 60)( 10, 59)( 11, 58)( 12, 57)( 13, 64)( 14, 63)( 15, 62)
( 16, 61)( 17, 68)( 18, 67)( 19, 66)( 20, 65)( 21, 72)( 22, 71)( 23, 70)
( 24, 69)( 25, 76)( 26, 75)( 27, 74)( 28, 73)( 29, 80)( 30, 79)( 31, 78)
( 32, 77)( 33, 84)( 34, 83)( 35, 82)( 36, 81)( 37, 88)( 38, 87)( 39, 86)
( 40, 85)( 41, 92)( 42, 91)( 43, 90)( 44, 89)( 45, 96)( 46, 95)( 47, 94)
( 48, 93)( 97,148)( 98,147)( 99,146)(100,145)(101,152)(102,151)(103,150)
(104,149)(105,156)(106,155)(107,154)(108,153)(109,160)(110,159)(111,158)
(112,157)(113,164)(114,163)(115,162)(116,161)(117,168)(118,167)(119,166)
(120,165)(121,172)(122,171)(123,170)(124,169)(125,176)(126,175)(127,174)
(128,173)(129,180)(130,179)(131,178)(132,177)(133,184)(134,183)(135,182)
(136,181)(137,188)(138,187)(139,186)(140,185)(141,192)(142,191)(143,190)
(144,189)(193,244)(194,243)(195,242)(196,241)(197,248)(198,247)(199,246)
(200,245)(201,252)(202,251)(203,250)(204,249)(205,256)(206,255)(207,254)
(208,253)(209,260)(210,259)(211,258)(212,257)(213,264)(214,263)(215,262)
(216,261)(217,268)(218,267)(219,266)(220,265)(221,272)(222,271)(223,270)
(224,269)(225,276)(226,275)(227,274)(228,273)(229,280)(230,279)(231,278)
(232,277)(233,284)(234,283)(235,282)(236,281)(237,288)(238,287)(239,286)
(240,285)(289,340)(290,339)(291,338)(292,337)(293,344)(294,343)(295,342)
(296,341)(297,348)(298,347)(299,346)(300,345)(301,352)(302,351)(303,350)
(304,349)(305,356)(306,355)(307,354)(308,353)(309,360)(310,359)(311,358)
(312,357)(313,364)(314,363)(315,362)(316,361)(317,368)(318,367)(319,366)
(320,365)(321,372)(322,371)(323,370)(324,369)(325,376)(326,375)(327,374)
(328,373)(329,380)(330,379)(331,378)(332,377)(333,384)(334,383)(335,382)
(336,381);
s2 := Sym(384)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 15, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 27, 28)( 29, 33)( 30, 34)( 31, 36)( 32, 35)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 51, 52)( 53, 57)( 54, 58)
( 55, 60)( 56, 59)( 63, 64)( 65, 69)( 66, 70)( 67, 72)( 68, 71)( 75, 76)
( 77, 81)( 78, 82)( 79, 84)( 80, 83)( 87, 88)( 89, 93)( 90, 94)( 91, 96)
( 92, 95)( 99,100)(101,105)(102,106)(103,108)(104,107)(111,112)(113,117)
(114,118)(115,120)(116,119)(123,124)(125,129)(126,130)(127,132)(128,131)
(135,136)(137,141)(138,142)(139,144)(140,143)(147,148)(149,153)(150,154)
(151,156)(152,155)(159,160)(161,165)(162,166)(163,168)(164,167)(171,172)
(173,177)(174,178)(175,180)(176,179)(183,184)(185,189)(186,190)(187,192)
(188,191)(193,205)(194,206)(195,208)(196,207)(197,213)(198,214)(199,216)
(200,215)(201,209)(202,210)(203,212)(204,211)(217,229)(218,230)(219,232)
(220,231)(221,237)(222,238)(223,240)(224,239)(225,233)(226,234)(227,236)
(228,235)(241,253)(242,254)(243,256)(244,255)(245,261)(246,262)(247,264)
(248,263)(249,257)(250,258)(251,260)(252,259)(265,277)(266,278)(267,280)
(268,279)(269,285)(270,286)(271,288)(272,287)(273,281)(274,282)(275,284)
(276,283)(289,301)(290,302)(291,304)(292,303)(293,309)(294,310)(295,312)
(296,311)(297,305)(298,306)(299,308)(300,307)(313,325)(314,326)(315,328)
(316,327)(317,333)(318,334)(319,336)(320,335)(321,329)(322,330)(323,332)
(324,331)(337,349)(338,350)(339,352)(340,351)(341,357)(342,358)(343,360)
(344,359)(345,353)(346,354)(347,356)(348,355)(361,373)(362,374)(363,376)
(364,375)(365,381)(366,382)(367,384)(368,383)(369,377)(370,378)(371,380)
(372,379);
s3 := Sym(384)!( 1,213)( 2,215)( 3,214)( 4,216)( 5,209)( 6,211)( 7,210)
( 8,212)( 9,205)( 10,207)( 11,206)( 12,208)( 13,201)( 14,203)( 15,202)
( 16,204)( 17,197)( 18,199)( 19,198)( 20,200)( 21,193)( 22,195)( 23,194)
( 24,196)( 25,237)( 26,239)( 27,238)( 28,240)( 29,233)( 30,235)( 31,234)
( 32,236)( 33,229)( 34,231)( 35,230)( 36,232)( 37,225)( 38,227)( 39,226)
( 40,228)( 41,221)( 42,223)( 43,222)( 44,224)( 45,217)( 46,219)( 47,218)
( 48,220)( 49,261)( 50,263)( 51,262)( 52,264)( 53,257)( 54,259)( 55,258)
( 56,260)( 57,253)( 58,255)( 59,254)( 60,256)( 61,249)( 62,251)( 63,250)
( 64,252)( 65,245)( 66,247)( 67,246)( 68,248)( 69,241)( 70,243)( 71,242)
( 72,244)( 73,285)( 74,287)( 75,286)( 76,288)( 77,281)( 78,283)( 79,282)
( 80,284)( 81,277)( 82,279)( 83,278)( 84,280)( 85,273)( 86,275)( 87,274)
( 88,276)( 89,269)( 90,271)( 91,270)( 92,272)( 93,265)( 94,267)( 95,266)
( 96,268)( 97,309)( 98,311)( 99,310)(100,312)(101,305)(102,307)(103,306)
(104,308)(105,301)(106,303)(107,302)(108,304)(109,297)(110,299)(111,298)
(112,300)(113,293)(114,295)(115,294)(116,296)(117,289)(118,291)(119,290)
(120,292)(121,333)(122,335)(123,334)(124,336)(125,329)(126,331)(127,330)
(128,332)(129,325)(130,327)(131,326)(132,328)(133,321)(134,323)(135,322)
(136,324)(137,317)(138,319)(139,318)(140,320)(141,313)(142,315)(143,314)
(144,316)(145,357)(146,359)(147,358)(148,360)(149,353)(150,355)(151,354)
(152,356)(153,349)(154,351)(155,350)(156,352)(157,345)(158,347)(159,346)
(160,348)(161,341)(162,343)(163,342)(164,344)(165,337)(166,339)(167,338)
(168,340)(169,381)(170,383)(171,382)(172,384)(173,377)(174,379)(175,378)
(176,380)(177,373)(178,375)(179,374)(180,376)(181,369)(182,371)(183,370)
(184,372)(185,365)(186,367)(187,366)(188,368)(189,361)(190,363)(191,362)
(192,364);
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
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