Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,54,4}

Atlas Canonical Name {2,54,4}*1728

Overview

Group
SmallGroup(1728,20782)
Rank
4
Schläfli Type
{2,54,4}
Vertices, edges, …
2, 108, 216, 8
Order of s0s1s2s3
54
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

18-fold

24-fold

36-fold

72-fold

108-fold

Covers minimal covers in bold

None in this atlas.

Representations

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