Part of the Atlas of Small Regular Polytopes

Polytope of Type {16,54}

Atlas Canonical Name {16,54}*1728

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,478)
Rank
3
Schläfli Type
{16,54}
Vertices, edges, …
16, 432, 54
Order of s0s1s2
432
Order of s0s1s2s1
2
Also known as
{16,54|2}. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

16-fold

18-fold

24-fold

27-fold

36-fold

48-fold

54-fold

72-fold

108-fold

144-fold

216-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

References

None.

to this polytope.

Twisty Puzzle