Polytope of Type {4,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*768a
if this polytope has a name.
Group : SmallGroup(768,1088927)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 16, 48, 16
Order of s0s1s2s3 : 4
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 : {4,6,6}*384, {2,6,12}*384a
4-fold quotients : {4,6,3}*192, {2,6,6}*192
8-fold quotients : {2,3,6}*96, {2,6,3}*96
12-fold quotients : {4,2,4}*64
16-fold quotients : {2,3,3}*48
24-fold quotients : {2,2,4}*32, {4,2,2}*32
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)( 7,199)( 8,200)
( 9,201)( 10,202)( 11,203)( 12,204)( 13,205)( 14,206)( 15,207)( 16,208)
( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)( 24,216)
( 25,217)( 26,218)( 27,219)( 28,220)( 29,221)( 30,222)( 31,223)( 32,224)
( 33,225)( 34,226)( 35,227)( 36,228)( 37,229)( 38,230)( 39,231)( 40,232)
( 41,233)( 42,234)( 43,235)( 44,236)( 45,237)( 46,238)( 47,239)( 48,240)
( 49,253)( 50,254)( 51,255)( 52,256)( 53,257)( 54,258)( 55,259)( 56,260)
( 57,261)( 58,262)( 59,263)( 60,264)( 61,241)( 62,242)( 63,243)( 64,244)
( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)( 72,252)
( 73,277)( 74,278)( 75,279)( 76,280)( 77,281)( 78,282)( 79,283)( 80,284)
( 81,285)( 82,286)( 83,287)( 84,288)( 85,265)( 86,266)( 87,267)( 88,268)
( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)( 96,276)
( 97,301)( 98,302)( 99,303)(100,304)(101,305)(102,306)(103,307)(104,308)
(105,309)(106,310)(107,311)(108,312)(109,289)(110,290)(111,291)(112,292)
(113,293)(114,294)(115,295)(116,296)(117,297)(118,298)(119,299)(120,300)
(121,325)(122,326)(123,327)(124,328)(125,329)(126,330)(127,331)(128,332)
(129,333)(130,334)(131,335)(132,336)(133,313)(134,314)(135,315)(136,316)
(137,317)(138,318)(139,319)(140,320)(141,321)(142,322)(143,323)(144,324)
(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,343)(152,344)
(153,345)(154,346)(155,347)(156,348)(157,349)(158,350)(159,351)(160,352)
(161,353)(162,354)(163,355)(164,356)(165,357)(166,358)(167,359)(168,360)
(169,361)(170,362)(171,363)(172,364)(173,365)(174,366)(175,367)(176,368)
(177,369)(178,370)(179,371)(180,372)(181,373)(182,374)(183,375)(184,376)
(185,377)(186,378)(187,379)(188,380)(189,381)(190,382)(191,383)(192,384);;
s1 := ( 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);;
s2 := ( 1,105)( 2,107)( 3,106)( 4,108)( 5,101)( 6,103)( 7,102)( 8,104)
( 9, 97)( 10, 99)( 11, 98)( 12,100)( 13,117)( 14,119)( 15,118)( 16,120)
( 17,113)( 18,115)( 19,114)( 20,116)( 21,109)( 22,111)( 23,110)( 24,112)
( 25,129)( 26,131)( 27,130)( 28,132)( 29,125)( 30,127)( 31,126)( 32,128)
( 33,121)( 34,123)( 35,122)( 36,124)( 37,141)( 38,143)( 39,142)( 40,144)
( 41,137)( 42,139)( 43,138)( 44,140)( 45,133)( 46,135)( 47,134)( 48,136)
( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)( 56,176)
( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)( 64,192)
( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)( 72,184)
( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)( 80,152)
( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)( 88,168)
( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)( 96,160)
(193,309)(194,311)(195,310)(196,312)(197,305)(198,307)(199,306)(200,308)
(201,301)(202,303)(203,302)(204,304)(205,297)(206,299)(207,298)(208,300)
(209,293)(210,295)(211,294)(212,296)(213,289)(214,291)(215,290)(216,292)
(217,333)(218,335)(219,334)(220,336)(221,329)(222,331)(223,330)(224,332)
(225,325)(226,327)(227,326)(228,328)(229,321)(230,323)(231,322)(232,324)
(233,317)(234,319)(235,318)(236,320)(237,313)(238,315)(239,314)(240,316)
(241,381)(242,383)(243,382)(244,384)(245,377)(246,379)(247,378)(248,380)
(249,373)(250,375)(251,374)(252,376)(253,369)(254,371)(255,370)(256,372)
(257,365)(258,367)(259,366)(260,368)(261,361)(262,363)(263,362)(264,364)
(265,357)(266,359)(267,358)(268,360)(269,353)(270,355)(271,354)(272,356)
(273,349)(274,351)(275,350)(276,352)(277,345)(278,347)(279,346)(280,348)
(281,341)(282,343)(283,342)(284,344)(285,337)(286,339)(287,338)(288,340);;
s3 := ( 1, 50)( 2, 49)( 3, 51)( 4, 52)( 5, 58)( 6, 57)( 7, 59)( 8, 60)
( 9, 54)( 10, 53)( 11, 55)( 12, 56)( 13, 62)( 14, 61)( 15, 63)( 16, 64)
( 17, 70)( 18, 69)( 19, 71)( 20, 72)( 21, 66)( 22, 65)( 23, 67)( 24, 68)
( 25, 74)( 26, 73)( 27, 75)( 28, 76)( 29, 82)( 30, 81)( 31, 83)( 32, 84)
( 33, 78)( 34, 77)( 35, 79)( 36, 80)( 37, 86)( 38, 85)( 39, 87)( 40, 88)
( 41, 94)( 42, 93)( 43, 95)( 44, 96)( 45, 90)( 46, 89)( 47, 91)( 48, 92)
( 97,146)( 98,145)( 99,147)(100,148)(101,154)(102,153)(103,155)(104,156)
(105,150)(106,149)(107,151)(108,152)(109,158)(110,157)(111,159)(112,160)
(113,166)(114,165)(115,167)(116,168)(117,162)(118,161)(119,163)(120,164)
(121,170)(122,169)(123,171)(124,172)(125,178)(126,177)(127,179)(128,180)
(129,174)(130,173)(131,175)(132,176)(133,182)(134,181)(135,183)(136,184)
(137,190)(138,189)(139,191)(140,192)(141,186)(142,185)(143,187)(144,188)
(193,254)(194,253)(195,255)(196,256)(197,262)(198,261)(199,263)(200,264)
(201,258)(202,257)(203,259)(204,260)(205,242)(206,241)(207,243)(208,244)
(209,250)(210,249)(211,251)(212,252)(213,246)(214,245)(215,247)(216,248)
(217,278)(218,277)(219,279)(220,280)(221,286)(222,285)(223,287)(224,288)
(225,282)(226,281)(227,283)(228,284)(229,266)(230,265)(231,267)(232,268)
(233,274)(234,273)(235,275)(236,276)(237,270)(238,269)(239,271)(240,272)
(289,350)(290,349)(291,351)(292,352)(293,358)(294,357)(295,359)(296,360)
(297,354)(298,353)(299,355)(300,356)(301,338)(302,337)(303,339)(304,340)
(305,346)(306,345)(307,347)(308,348)(309,342)(310,341)(311,343)(312,344)
(313,374)(314,373)(315,375)(316,376)(317,382)(318,381)(319,383)(320,384)
(321,378)(322,377)(323,379)(324,380)(325,362)(326,361)(327,363)(328,364)
(329,370)(330,369)(331,371)(332,372)(333,366)(334,365)(335,367)(336,368);;
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, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(384)!( 1,193)( 2,194)( 3,195)( 4,196)( 5,197)( 6,198)( 7,199)
( 8,200)( 9,201)( 10,202)( 11,203)( 12,204)( 13,205)( 14,206)( 15,207)
( 16,208)( 17,209)( 18,210)( 19,211)( 20,212)( 21,213)( 22,214)( 23,215)
( 24,216)( 25,217)( 26,218)( 27,219)( 28,220)( 29,221)( 30,222)( 31,223)
( 32,224)( 33,225)( 34,226)( 35,227)( 36,228)( 37,229)( 38,230)( 39,231)
( 40,232)( 41,233)( 42,234)( 43,235)( 44,236)( 45,237)( 46,238)( 47,239)
( 48,240)( 49,253)( 50,254)( 51,255)( 52,256)( 53,257)( 54,258)( 55,259)
( 56,260)( 57,261)( 58,262)( 59,263)( 60,264)( 61,241)( 62,242)( 63,243)
( 64,244)( 65,245)( 66,246)( 67,247)( 68,248)( 69,249)( 70,250)( 71,251)
( 72,252)( 73,277)( 74,278)( 75,279)( 76,280)( 77,281)( 78,282)( 79,283)
( 80,284)( 81,285)( 82,286)( 83,287)( 84,288)( 85,265)( 86,266)( 87,267)
( 88,268)( 89,269)( 90,270)( 91,271)( 92,272)( 93,273)( 94,274)( 95,275)
( 96,276)( 97,301)( 98,302)( 99,303)(100,304)(101,305)(102,306)(103,307)
(104,308)(105,309)(106,310)(107,311)(108,312)(109,289)(110,290)(111,291)
(112,292)(113,293)(114,294)(115,295)(116,296)(117,297)(118,298)(119,299)
(120,300)(121,325)(122,326)(123,327)(124,328)(125,329)(126,330)(127,331)
(128,332)(129,333)(130,334)(131,335)(132,336)(133,313)(134,314)(135,315)
(136,316)(137,317)(138,318)(139,319)(140,320)(141,321)(142,322)(143,323)
(144,324)(145,337)(146,338)(147,339)(148,340)(149,341)(150,342)(151,343)
(152,344)(153,345)(154,346)(155,347)(156,348)(157,349)(158,350)(159,351)
(160,352)(161,353)(162,354)(163,355)(164,356)(165,357)(166,358)(167,359)
(168,360)(169,361)(170,362)(171,363)(172,364)(173,365)(174,366)(175,367)
(176,368)(177,369)(178,370)(179,371)(180,372)(181,373)(182,374)(183,375)
(184,376)(185,377)(186,378)(187,379)(188,380)(189,381)(190,382)(191,383)
(192,384);
s1 := 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);
s2 := Sym(384)!( 1,105)( 2,107)( 3,106)( 4,108)( 5,101)( 6,103)( 7,102)
( 8,104)( 9, 97)( 10, 99)( 11, 98)( 12,100)( 13,117)( 14,119)( 15,118)
( 16,120)( 17,113)( 18,115)( 19,114)( 20,116)( 21,109)( 22,111)( 23,110)
( 24,112)( 25,129)( 26,131)( 27,130)( 28,132)( 29,125)( 30,127)( 31,126)
( 32,128)( 33,121)( 34,123)( 35,122)( 36,124)( 37,141)( 38,143)( 39,142)
( 40,144)( 41,137)( 42,139)( 43,138)( 44,140)( 45,133)( 46,135)( 47,134)
( 48,136)( 49,177)( 50,179)( 51,178)( 52,180)( 53,173)( 54,175)( 55,174)
( 56,176)( 57,169)( 58,171)( 59,170)( 60,172)( 61,189)( 62,191)( 63,190)
( 64,192)( 65,185)( 66,187)( 67,186)( 68,188)( 69,181)( 70,183)( 71,182)
( 72,184)( 73,153)( 74,155)( 75,154)( 76,156)( 77,149)( 78,151)( 79,150)
( 80,152)( 81,145)( 82,147)( 83,146)( 84,148)( 85,165)( 86,167)( 87,166)
( 88,168)( 89,161)( 90,163)( 91,162)( 92,164)( 93,157)( 94,159)( 95,158)
( 96,160)(193,309)(194,311)(195,310)(196,312)(197,305)(198,307)(199,306)
(200,308)(201,301)(202,303)(203,302)(204,304)(205,297)(206,299)(207,298)
(208,300)(209,293)(210,295)(211,294)(212,296)(213,289)(214,291)(215,290)
(216,292)(217,333)(218,335)(219,334)(220,336)(221,329)(222,331)(223,330)
(224,332)(225,325)(226,327)(227,326)(228,328)(229,321)(230,323)(231,322)
(232,324)(233,317)(234,319)(235,318)(236,320)(237,313)(238,315)(239,314)
(240,316)(241,381)(242,383)(243,382)(244,384)(245,377)(246,379)(247,378)
(248,380)(249,373)(250,375)(251,374)(252,376)(253,369)(254,371)(255,370)
(256,372)(257,365)(258,367)(259,366)(260,368)(261,361)(262,363)(263,362)
(264,364)(265,357)(266,359)(267,358)(268,360)(269,353)(270,355)(271,354)
(272,356)(273,349)(274,351)(275,350)(276,352)(277,345)(278,347)(279,346)
(280,348)(281,341)(282,343)(283,342)(284,344)(285,337)(286,339)(287,338)
(288,340);
s3 := Sym(384)!( 1, 50)( 2, 49)( 3, 51)( 4, 52)( 5, 58)( 6, 57)( 7, 59)
( 8, 60)( 9, 54)( 10, 53)( 11, 55)( 12, 56)( 13, 62)( 14, 61)( 15, 63)
( 16, 64)( 17, 70)( 18, 69)( 19, 71)( 20, 72)( 21, 66)( 22, 65)( 23, 67)
( 24, 68)( 25, 74)( 26, 73)( 27, 75)( 28, 76)( 29, 82)( 30, 81)( 31, 83)
( 32, 84)( 33, 78)( 34, 77)( 35, 79)( 36, 80)( 37, 86)( 38, 85)( 39, 87)
( 40, 88)( 41, 94)( 42, 93)( 43, 95)( 44, 96)( 45, 90)( 46, 89)( 47, 91)
( 48, 92)( 97,146)( 98,145)( 99,147)(100,148)(101,154)(102,153)(103,155)
(104,156)(105,150)(106,149)(107,151)(108,152)(109,158)(110,157)(111,159)
(112,160)(113,166)(114,165)(115,167)(116,168)(117,162)(118,161)(119,163)
(120,164)(121,170)(122,169)(123,171)(124,172)(125,178)(126,177)(127,179)
(128,180)(129,174)(130,173)(131,175)(132,176)(133,182)(134,181)(135,183)
(136,184)(137,190)(138,189)(139,191)(140,192)(141,186)(142,185)(143,187)
(144,188)(193,254)(194,253)(195,255)(196,256)(197,262)(198,261)(199,263)
(200,264)(201,258)(202,257)(203,259)(204,260)(205,242)(206,241)(207,243)
(208,244)(209,250)(210,249)(211,251)(212,252)(213,246)(214,245)(215,247)
(216,248)(217,278)(218,277)(219,279)(220,280)(221,286)(222,285)(223,287)
(224,288)(225,282)(226,281)(227,283)(228,284)(229,266)(230,265)(231,267)
(232,268)(233,274)(234,273)(235,275)(236,276)(237,270)(238,269)(239,271)
(240,272)(289,350)(290,349)(291,351)(292,352)(293,358)(294,357)(295,359)
(296,360)(297,354)(298,353)(299,355)(300,356)(301,338)(302,337)(303,339)
(304,340)(305,346)(306,345)(307,347)(308,348)(309,342)(310,341)(311,343)
(312,344)(313,374)(314,373)(315,375)(316,376)(317,382)(318,381)(319,383)
(320,384)(321,378)(322,377)(323,379)(324,380)(325,362)(326,361)(327,363)
(328,364)(329,370)(330,369)(331,371)(332,372)(333,366)(334,365)(335,367)
(336,368);
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,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2 >;
References : None.
to this polytope