Polytope of Type {12,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,12}*1728g
if this polytope has a name.
Group : SmallGroup(1728,38764)
Rank : 4
Schlafli Type : {12,6,12}
Number of vertices, edges, etc : 12, 36, 36, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
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 : {6,6,12}*864g, {12,6,6}*864g
3-fold quotients : {4,6,12}*576c, {12,6,4}*576c
4-fold quotients : {6,6,6}*432f
6-fold quotients : {4,6,6}*288b, {6,6,4}*288b, {2,6,12}*288c, {12,6,2}*288c
8-fold quotients : {6,3,6}*216
9-fold quotients : {4,6,4}*192a
12-fold quotients : {2,6,6}*144c, {6,6,2}*144b
18-fold quotients : {2,6,4}*96a, {4,6,2}*96a
24-fold quotients : {2,3,6}*72, {6,3,2}*72
27-fold quotients : {4,2,4}*64
36-fold quotients : {2,6,2}*48
54-fold quotients : {2,2,4}*32, {4,2,2}*32
72-fold quotients : {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)
(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)
(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213)(217,244)(218,245)(219,246)(220,250)(221,251)(222,252)
(223,247)(224,248)(225,249)(226,253)(227,254)(228,255)(229,259)(230,260)
(231,261)(232,256)(233,257)(234,258)(235,262)(236,263)(237,264)(238,268)
(239,269)(240,270)(241,265)(242,266)(243,267)(271,298)(272,299)(273,300)
(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)(281,308)
(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)(289,316)
(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)(297,321)
(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)(347,350)
(348,351)(355,358)(356,359)(357,360)(364,367)(365,368)(366,369)(373,376)
(374,377)(375,378)(382,385)(383,386)(384,387)(391,394)(392,395)(393,396)
(400,403)(401,404)(402,405)(409,412)(410,413)(411,414)(418,421)(419,422)
(420,423)(427,430)(428,431)(429,432);;
s1 := ( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,223)( 8,225)
( 9,224)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)( 16,241)
( 17,243)( 18,242)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)( 24,227)
( 25,232)( 26,234)( 27,233)( 28,247)( 29,249)( 30,248)( 31,244)( 32,246)
( 33,245)( 34,250)( 35,252)( 36,251)( 37,265)( 38,267)( 39,266)( 40,262)
( 41,264)( 42,263)( 43,268)( 44,270)( 45,269)( 46,256)( 47,258)( 48,257)
( 49,253)( 50,255)( 51,254)( 52,259)( 53,261)( 54,260)( 55,274)( 56,276)
( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)( 64,292)
( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,295)( 71,297)( 72,296)
( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,286)( 80,288)
( 81,287)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)( 88,304)
( 89,306)( 90,305)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)( 96,317)
( 97,322)( 98,324)( 99,323)(100,310)(101,312)(102,311)(103,307)(104,309)
(105,308)(106,313)(107,315)(108,314)(109,328)(110,330)(111,329)(112,325)
(113,327)(114,326)(115,331)(116,333)(117,332)(118,346)(119,348)(120,347)
(121,343)(122,345)(123,344)(124,349)(125,351)(126,350)(127,337)(128,339)
(129,338)(130,334)(131,336)(132,335)(133,340)(134,342)(135,341)(136,355)
(137,357)(138,356)(139,352)(140,354)(141,353)(142,358)(143,360)(144,359)
(145,373)(146,375)(147,374)(148,370)(149,372)(150,371)(151,376)(152,378)
(153,377)(154,364)(155,366)(156,365)(157,361)(158,363)(159,362)(160,367)
(161,369)(162,368)(163,382)(164,384)(165,383)(166,379)(167,381)(168,380)
(169,385)(170,387)(171,386)(172,400)(173,402)(174,401)(175,397)(176,399)
(177,398)(178,403)(179,405)(180,404)(181,391)(182,393)(183,392)(184,388)
(185,390)(186,389)(187,394)(188,396)(189,395)(190,409)(191,411)(192,410)
(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,427)(200,429)
(201,428)(202,424)(203,426)(204,425)(205,430)(206,432)(207,431)(208,418)
(209,420)(210,419)(211,415)(212,417)(213,416)(214,421)(215,423)(216,422);;
s2 := ( 1,335)( 2,334)( 3,336)( 4,341)( 5,340)( 6,342)( 7,338)( 8,337)
( 9,339)( 10,326)( 11,325)( 12,327)( 13,332)( 14,331)( 15,333)( 16,329)
( 17,328)( 18,330)( 19,344)( 20,343)( 21,345)( 22,350)( 23,349)( 24,351)
( 25,347)( 26,346)( 27,348)( 28,362)( 29,361)( 30,363)( 31,368)( 32,367)
( 33,369)( 34,365)( 35,364)( 36,366)( 37,353)( 38,352)( 39,354)( 40,359)
( 41,358)( 42,360)( 43,356)( 44,355)( 45,357)( 46,371)( 47,370)( 48,372)
( 49,377)( 50,376)( 51,378)( 52,374)( 53,373)( 54,375)( 55,389)( 56,388)
( 57,390)( 58,395)( 59,394)( 60,396)( 61,392)( 62,391)( 63,393)( 64,380)
( 65,379)( 66,381)( 67,386)( 68,385)( 69,387)( 70,383)( 71,382)( 72,384)
( 73,398)( 74,397)( 75,399)( 76,404)( 77,403)( 78,405)( 79,401)( 80,400)
( 81,402)( 82,416)( 83,415)( 84,417)( 85,422)( 86,421)( 87,423)( 88,419)
( 89,418)( 90,420)( 91,407)( 92,406)( 93,408)( 94,413)( 95,412)( 96,414)
( 97,410)( 98,409)( 99,411)(100,425)(101,424)(102,426)(103,431)(104,430)
(105,432)(106,428)(107,427)(108,429)(109,227)(110,226)(111,228)(112,233)
(113,232)(114,234)(115,230)(116,229)(117,231)(118,218)(119,217)(120,219)
(121,224)(122,223)(123,225)(124,221)(125,220)(126,222)(127,236)(128,235)
(129,237)(130,242)(131,241)(132,243)(133,239)(134,238)(135,240)(136,254)
(137,253)(138,255)(139,260)(140,259)(141,261)(142,257)(143,256)(144,258)
(145,245)(146,244)(147,246)(148,251)(149,250)(150,252)(151,248)(152,247)
(153,249)(154,263)(155,262)(156,264)(157,269)(158,268)(159,270)(160,266)
(161,265)(162,267)(163,281)(164,280)(165,282)(166,287)(167,286)(168,288)
(169,284)(170,283)(171,285)(172,272)(173,271)(174,273)(175,278)(176,277)
(177,279)(178,275)(179,274)(180,276)(181,290)(182,289)(183,291)(184,296)
(185,295)(186,297)(187,293)(188,292)(189,294)(190,308)(191,307)(192,309)
(193,314)(194,313)(195,315)(196,311)(197,310)(198,312)(199,299)(200,298)
(201,300)(202,305)(203,304)(204,306)(205,302)(206,301)(207,303)(208,317)
(209,316)(210,318)(211,323)(212,322)(213,324)(214,320)(215,319)(216,321);;
s3 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(109,163)(110,164)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)(120,183)
(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)(128,173)
(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)
(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)(160,205)
(161,206)(162,207)(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)
(232,241)(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)
(258,267)(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)
(284,293)(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)
(310,319)(311,320)(312,321)(313,322)(314,323)(315,324)(325,379)(326,380)
(327,381)(328,382)(329,383)(330,384)(331,385)(332,386)(333,387)(334,397)
(335,398)(336,399)(337,400)(338,401)(339,402)(340,403)(341,404)(342,405)
(343,388)(344,389)(345,390)(346,391)(347,392)(348,393)(349,394)(350,395)
(351,396)(352,406)(353,407)(354,408)(355,409)(356,410)(357,411)(358,412)
(359,413)(360,414)(361,424)(362,425)(363,426)(364,427)(365,428)(366,429)
(367,430)(368,431)(369,432)(370,415)(371,416)(372,417)(373,418)(374,419)
(375,420)(376,421)(377,422)(378,423);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213)(217,244)(218,245)(219,246)(220,250)(221,251)
(222,252)(223,247)(224,248)(225,249)(226,253)(227,254)(228,255)(229,259)
(230,260)(231,261)(232,256)(233,257)(234,258)(235,262)(236,263)(237,264)
(238,268)(239,269)(240,270)(241,265)(242,266)(243,267)(271,298)(272,299)
(273,300)(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)
(281,308)(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)
(289,316)(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)
(297,321)(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)
(347,350)(348,351)(355,358)(356,359)(357,360)(364,367)(365,368)(366,369)
(373,376)(374,377)(375,378)(382,385)(383,386)(384,387)(391,394)(392,395)
(393,396)(400,403)(401,404)(402,405)(409,412)(410,413)(411,414)(418,421)
(419,422)(420,423)(427,430)(428,431)(429,432);
s1 := Sym(432)!( 1,220)( 2,222)( 3,221)( 4,217)( 5,219)( 6,218)( 7,223)
( 8,225)( 9,224)( 10,238)( 11,240)( 12,239)( 13,235)( 14,237)( 15,236)
( 16,241)( 17,243)( 18,242)( 19,229)( 20,231)( 21,230)( 22,226)( 23,228)
( 24,227)( 25,232)( 26,234)( 27,233)( 28,247)( 29,249)( 30,248)( 31,244)
( 32,246)( 33,245)( 34,250)( 35,252)( 36,251)( 37,265)( 38,267)( 39,266)
( 40,262)( 41,264)( 42,263)( 43,268)( 44,270)( 45,269)( 46,256)( 47,258)
( 48,257)( 49,253)( 50,255)( 51,254)( 52,259)( 53,261)( 54,260)( 55,274)
( 56,276)( 57,275)( 58,271)( 59,273)( 60,272)( 61,277)( 62,279)( 63,278)
( 64,292)( 65,294)( 66,293)( 67,289)( 68,291)( 69,290)( 70,295)( 71,297)
( 72,296)( 73,283)( 74,285)( 75,284)( 76,280)( 77,282)( 78,281)( 79,286)
( 80,288)( 81,287)( 82,301)( 83,303)( 84,302)( 85,298)( 86,300)( 87,299)
( 88,304)( 89,306)( 90,305)( 91,319)( 92,321)( 93,320)( 94,316)( 95,318)
( 96,317)( 97,322)( 98,324)( 99,323)(100,310)(101,312)(102,311)(103,307)
(104,309)(105,308)(106,313)(107,315)(108,314)(109,328)(110,330)(111,329)
(112,325)(113,327)(114,326)(115,331)(116,333)(117,332)(118,346)(119,348)
(120,347)(121,343)(122,345)(123,344)(124,349)(125,351)(126,350)(127,337)
(128,339)(129,338)(130,334)(131,336)(132,335)(133,340)(134,342)(135,341)
(136,355)(137,357)(138,356)(139,352)(140,354)(141,353)(142,358)(143,360)
(144,359)(145,373)(146,375)(147,374)(148,370)(149,372)(150,371)(151,376)
(152,378)(153,377)(154,364)(155,366)(156,365)(157,361)(158,363)(159,362)
(160,367)(161,369)(162,368)(163,382)(164,384)(165,383)(166,379)(167,381)
(168,380)(169,385)(170,387)(171,386)(172,400)(173,402)(174,401)(175,397)
(176,399)(177,398)(178,403)(179,405)(180,404)(181,391)(182,393)(183,392)
(184,388)(185,390)(186,389)(187,394)(188,396)(189,395)(190,409)(191,411)
(192,410)(193,406)(194,408)(195,407)(196,412)(197,414)(198,413)(199,427)
(200,429)(201,428)(202,424)(203,426)(204,425)(205,430)(206,432)(207,431)
(208,418)(209,420)(210,419)(211,415)(212,417)(213,416)(214,421)(215,423)
(216,422);
s2 := Sym(432)!( 1,335)( 2,334)( 3,336)( 4,341)( 5,340)( 6,342)( 7,338)
( 8,337)( 9,339)( 10,326)( 11,325)( 12,327)( 13,332)( 14,331)( 15,333)
( 16,329)( 17,328)( 18,330)( 19,344)( 20,343)( 21,345)( 22,350)( 23,349)
( 24,351)( 25,347)( 26,346)( 27,348)( 28,362)( 29,361)( 30,363)( 31,368)
( 32,367)( 33,369)( 34,365)( 35,364)( 36,366)( 37,353)( 38,352)( 39,354)
( 40,359)( 41,358)( 42,360)( 43,356)( 44,355)( 45,357)( 46,371)( 47,370)
( 48,372)( 49,377)( 50,376)( 51,378)( 52,374)( 53,373)( 54,375)( 55,389)
( 56,388)( 57,390)( 58,395)( 59,394)( 60,396)( 61,392)( 62,391)( 63,393)
( 64,380)( 65,379)( 66,381)( 67,386)( 68,385)( 69,387)( 70,383)( 71,382)
( 72,384)( 73,398)( 74,397)( 75,399)( 76,404)( 77,403)( 78,405)( 79,401)
( 80,400)( 81,402)( 82,416)( 83,415)( 84,417)( 85,422)( 86,421)( 87,423)
( 88,419)( 89,418)( 90,420)( 91,407)( 92,406)( 93,408)( 94,413)( 95,412)
( 96,414)( 97,410)( 98,409)( 99,411)(100,425)(101,424)(102,426)(103,431)
(104,430)(105,432)(106,428)(107,427)(108,429)(109,227)(110,226)(111,228)
(112,233)(113,232)(114,234)(115,230)(116,229)(117,231)(118,218)(119,217)
(120,219)(121,224)(122,223)(123,225)(124,221)(125,220)(126,222)(127,236)
(128,235)(129,237)(130,242)(131,241)(132,243)(133,239)(134,238)(135,240)
(136,254)(137,253)(138,255)(139,260)(140,259)(141,261)(142,257)(143,256)
(144,258)(145,245)(146,244)(147,246)(148,251)(149,250)(150,252)(151,248)
(152,247)(153,249)(154,263)(155,262)(156,264)(157,269)(158,268)(159,270)
(160,266)(161,265)(162,267)(163,281)(164,280)(165,282)(166,287)(167,286)
(168,288)(169,284)(170,283)(171,285)(172,272)(173,271)(174,273)(175,278)
(176,277)(177,279)(178,275)(179,274)(180,276)(181,290)(182,289)(183,291)
(184,296)(185,295)(186,297)(187,293)(188,292)(189,294)(190,308)(191,307)
(192,309)(193,314)(194,313)(195,315)(196,311)(197,310)(198,312)(199,299)
(200,298)(201,300)(202,305)(203,304)(204,306)(205,302)(206,301)(207,303)
(208,317)(209,316)(210,318)(211,323)(212,322)(213,324)(214,320)(215,319)
(216,321);
s3 := Sym(432)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,163)(110,164)(111,165)
(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,181)(119,182)
(120,183)(121,184)(122,185)(123,186)(124,187)(125,188)(126,189)(127,172)
(128,173)(129,174)(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)
(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)
(144,198)(145,208)(146,209)(147,210)(148,211)(149,212)(150,213)(151,214)
(152,215)(153,216)(154,199)(155,200)(156,201)(157,202)(158,203)(159,204)
(160,205)(161,206)(162,207)(226,235)(227,236)(228,237)(229,238)(230,239)
(231,240)(232,241)(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)
(257,266)(258,267)(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)
(283,292)(284,293)(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)
(309,318)(310,319)(311,320)(312,321)(313,322)(314,323)(315,324)(325,379)
(326,380)(327,381)(328,382)(329,383)(330,384)(331,385)(332,386)(333,387)
(334,397)(335,398)(336,399)(337,400)(338,401)(339,402)(340,403)(341,404)
(342,405)(343,388)(344,389)(345,390)(346,391)(347,392)(348,393)(349,394)
(350,395)(351,396)(352,406)(353,407)(354,408)(355,409)(356,410)(357,411)
(358,412)(359,413)(360,414)(361,424)(362,425)(363,426)(364,427)(365,428)
(366,429)(367,430)(368,431)(369,432)(370,415)(371,416)(372,417)(373,418)
(374,419)(375,420)(376,421)(377,422)(378,423);
poly := sub<Sym(432)|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,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope