Polytope of Type {8,27,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,27,2}*1728
if this polytope has a name.
Group : SmallGroup(1728,11367)
Rank : 4
Schlafli Type : {8,27,2}
Number of vertices, edges, etc : 16, 216, 54, 2
Order of s₀s₁s₂s₃ : 108
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {4,27,2}*864
   3-fold quotients : {8,9,2}*576
   4-fold quotients : {4,27,2}*432
   6-fold quotients : {4,9,2}*288
   8-fold quotients : {2,27,2}*216
   9-fold quotients : {8,3,2}*192
   12-fold quotients : {4,9,2}*144
   18-fold quotients : {4,3,2}*96
   24-fold quotients : {2,9,2}*72
   36-fold quotients : {4,3,2}*48
   72-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.

Permutation Representation (GAP) :

s0 := (  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)( 16,228)( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)( 24,236)( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)( 32,244)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)( 48,260)( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)( 56,268)( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)( 64,276)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)( 88,300)( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)( 96,308)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)(112,324)(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)(120,332)(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)(128,340)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)(144,356)(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)(160,372)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)(176,388)(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)(184,396)(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)(192,404)(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)(200,412)(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)(208,420)(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)(216,428);;
s1 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)( 80,215)( 81,201)( 82,202)( 83,206)( 84,205)( 85,204)( 86,203)( 87,208)( 88,207)( 89,193)( 90,194)( 91,198)( 92,197)( 93,196)( 94,195)( 95,200)( 96,199)( 97,185)( 98,186)( 99,190)(100,189)(101,188)(102,187)(103,192)(104,191)(105,177)(106,178)(107,182)(108,181)(109,180)(110,179)(111,184)(112,183)(113,169)(114,170)(115,174)(116,173)(117,172)(118,171)(119,176)(120,175)(121,161)(122,162)(123,166)(124,165)(125,164)(126,163)(127,168)(128,167)(129,153)(130,154)(131,158)(132,157)(133,156)(134,155)(135,160)(136,159)(137,145)(138,146)(139,150)(140,149)(141,148)(142,147)(143,152)(144,151)(217,218)(219,221)(220,222)(225,234)(226,233)(227,237)(228,238)(229,235)(230,236)(231,239)(232,240)(241,282)(242,281)(243,285)(244,286)(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)(251,277)(252,278)(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)(259,269)(260,270)(261,267)(262,268)(263,271)(264,272)(289,426)(290,425)(291,429)(292,430)(293,427)(294,428)(295,431)(296,432)(297,418)(298,417)(299,421)(300,422)(301,419)(302,420)(303,423)(304,424)(305,410)(306,409)(307,413)(308,414)(309,411)(310,412)(311,415)(312,416)(313,402)(314,401)(315,405)(316,406)(317,403)(318,404)(319,407)(320,408)(321,394)(322,393)(323,397)(324,398)(325,395)(326,396)(327,399)(328,400)(329,386)(330,385)(331,389)(332,390)(333,387)(334,388)(335,391)(336,392)(337,378)(338,377)(339,381)(340,382)(341,379)(342,380)(343,383)(344,384)(345,370)(346,369)(347,373)(348,374)(349,371)(350,372)(351,375)(352,376)(353,362)(354,361)(355,365)(356,366)(357,363)(358,364)(359,367)(360,368);;
s2 := (  1,145)(  2,146)(  3,151)(  4,152)(  5,150)(  6,149)(  7,147)(  8,148)(  9,161)( 10,162)( 11,167)( 12,168)( 13,166)( 14,165)( 15,163)( 16,164)( 17,153)( 18,154)( 19,159)( 20,160)( 21,158)( 22,157)( 23,155)( 24,156)( 25,209)( 26,210)( 27,215)( 28,216)( 29,214)( 30,213)( 31,211)( 32,212)( 33,201)( 34,202)( 35,207)( 36,208)( 37,206)( 38,205)( 39,203)( 40,204)( 41,193)( 42,194)( 43,199)( 44,200)( 45,198)( 46,197)( 47,195)( 48,196)( 49,185)( 50,186)( 51,191)( 52,192)( 53,190)( 54,189)( 55,187)( 56,188)( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)( 64,180)( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)( 72,172)( 75, 79)( 76, 80)( 77, 78)( 81, 89)( 82, 90)( 83, 95)( 84, 96)( 85, 94)( 86, 93)( 87, 91)( 88, 92)( 97,137)( 98,138)( 99,143)(100,144)(101,142)(102,141)(103,139)(104,140)(105,129)(106,130)(107,135)(108,136)(109,134)(110,133)(111,131)(112,132)(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)(120,124)(217,362)(218,361)(219,368)(220,367)(221,365)(222,366)(223,364)(224,363)(225,378)(226,377)(227,384)(228,383)(229,381)(230,382)(231,380)(232,379)(233,370)(234,369)(235,376)(236,375)(237,373)(238,374)(239,372)(240,371)(241,426)(242,425)(243,432)(244,431)(245,429)(246,430)(247,428)(248,427)(249,418)(250,417)(251,424)(252,423)(253,421)(254,422)(255,420)(256,419)(257,410)(258,409)(259,416)(260,415)(261,413)(262,414)(263,412)(264,411)(265,402)(266,401)(267,408)(268,407)(269,405)(270,406)(271,404)(272,403)(273,394)(274,393)(275,400)(276,399)(277,397)(278,398)(279,396)(280,395)(281,386)(282,385)(283,392)(284,391)(285,389)(286,390)(287,388)(288,387)(289,290)(291,296)(292,295)(297,306)(298,305)(299,312)(300,311)(301,309)(302,310)(303,308)(304,307)(313,354)(314,353)(315,360)(316,359)(317,357)(318,358)(319,356)(320,355)(321,346)(322,345)(323,352)(324,351)(325,349)(326,350)(327,348)(328,347)(329,338)(330,337)(331,344)(332,343)(333,341)(334,342)(335,340)(336,339);;
s3 := (433,434);;
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*s3*s2*s3, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
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*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*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(434)!(  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)( 16,228)( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)( 24,236)( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)( 32,244)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)( 48,260)( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)( 56,268)( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)( 64,276)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)( 88,300)( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)( 96,308)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)(112,324)(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)(120,332)(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)(128,340)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)(144,356)(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)(160,372)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)(176,388)(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)(184,396)(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)(192,404)(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)(200,412)(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)(208,420)(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)(216,428);
s1 := Sym(434)!(  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)( 78,211)( 79,216)( 80,215)( 81,201)( 82,202)( 83,206)( 84,205)( 85,204)( 86,203)( 87,208)( 88,207)( 89,193)( 90,194)( 91,198)( 92,197)( 93,196)( 94,195)( 95,200)( 96,199)( 97,185)( 98,186)( 99,190)(100,189)(101,188)(102,187)(103,192)(104,191)(105,177)(106,178)(107,182)(108,181)(109,180)(110,179)(111,184)(112,183)(113,169)(114,170)(115,174)(116,173)(117,172)(118,171)(119,176)(120,175)(121,161)(122,162)(123,166)(124,165)(125,164)(126,163)(127,168)(128,167)(129,153)(130,154)(131,158)(132,157)(133,156)(134,155)(135,160)(136,159)(137,145)(138,146)(139,150)(140,149)(141,148)(142,147)(143,152)(144,151)(217,218)(219,221)(220,222)(225,234)(226,233)(227,237)(228,238)(229,235)(230,236)(231,239)(232,240)(241,282)(242,281)(243,285)(244,286)(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)(251,277)(252,278)(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)(259,269)(260,270)(261,267)(262,268)(263,271)(264,272)(289,426)(290,425)(291,429)(292,430)(293,427)(294,428)(295,431)(296,432)(297,418)(298,417)(299,421)(300,422)(301,419)(302,420)(303,423)(304,424)(305,410)(306,409)(307,413)(308,414)(309,411)(310,412)(311,415)(312,416)(313,402)(314,401)(315,405)(316,406)(317,403)(318,404)(319,407)(320,408)(321,394)(322,393)(323,397)(324,398)(325,395)(326,396)(327,399)(328,400)(329,386)(330,385)(331,389)(332,390)(333,387)(334,388)(335,391)(336,392)(337,378)(338,377)(339,381)(340,382)(341,379)(342,380)(343,383)(344,384)(345,370)(346,369)(347,373)(348,374)(349,371)(350,372)(351,375)(352,376)(353,362)(354,361)(355,365)(356,366)(357,363)(358,364)(359,367)(360,368);
s2 := Sym(434)!(  1,145)(  2,146)(  3,151)(  4,152)(  5,150)(  6,149)(  7,147)(  8,148)(  9,161)( 10,162)( 11,167)( 12,168)( 13,166)( 14,165)( 15,163)( 16,164)( 17,153)( 18,154)( 19,159)( 20,160)( 21,158)( 22,157)( 23,155)( 24,156)( 25,209)( 26,210)( 27,215)( 28,216)( 29,214)( 30,213)( 31,211)( 32,212)( 33,201)( 34,202)( 35,207)( 36,208)( 37,206)( 38,205)( 39,203)( 40,204)( 41,193)( 42,194)( 43,199)( 44,200)( 45,198)( 46,197)( 47,195)( 48,196)( 49,185)( 50,186)( 51,191)( 52,192)( 53,190)( 54,189)( 55,187)( 56,188)( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)( 64,180)( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)( 72,172)( 75, 79)( 76, 80)( 77, 78)( 81, 89)( 82, 90)( 83, 95)( 84, 96)( 85, 94)( 86, 93)( 87, 91)( 88, 92)( 97,137)( 98,138)( 99,143)(100,144)(101,142)(102,141)(103,139)(104,140)(105,129)(106,130)(107,135)(108,136)(109,134)(110,133)(111,131)(112,132)(113,121)(114,122)(115,127)(116,128)(117,126)(118,125)(119,123)(120,124)(217,362)(218,361)(219,368)(220,367)(221,365)(222,366)(223,364)(224,363)(225,378)(226,377)(227,384)(228,383)(229,381)(230,382)(231,380)(232,379)(233,370)(234,369)(235,376)(236,375)(237,373)(238,374)(239,372)(240,371)(241,426)(242,425)(243,432)(244,431)(245,429)(246,430)(247,428)(248,427)(249,418)(250,417)(251,424)(252,423)(253,421)(254,422)(255,420)(256,419)(257,410)(258,409)(259,416)(260,415)(261,413)(262,414)(263,412)(264,411)(265,402)(266,401)(267,408)(268,407)(269,405)(270,406)(271,404)(272,403)(273,394)(274,393)(275,400)(276,399)(277,397)(278,398)(279,396)(280,395)(281,386)(282,385)(283,392)(284,391)(285,389)(286,390)(287,388)(288,387)(289,290)(291,296)(292,295)(297,306)(298,305)(299,312)(300,311)(301,309)(302,310)(303,308)(304,307)(313,354)(314,353)(315,360)(316,359)(317,357)(318,358)(319,356)(320,355)(321,346)(322,345)(323,352)(324,351)(325,349)(326,350)(327,348)(328,347)(329,338)(330,337)(331,344)(332,343)(333,341)(334,342)(335,340)(336,339);
s3 := Sym(434)!(433,434);
poly := sub<Sym(434)|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*s3*s2*s3, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
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*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*s1*s2 >;

to this polytope