Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,6,16}

Atlas Canonical Name {3,6,16}*1728a

Overview

Group
SmallGroup(1728,5302)
Rank
4
Schläfli Type
{3,6,16}
Vertices, edges, …
9, 27, 144, 16
Order of s0s1s2s3
48
Order of s0s1s2s3s2s1
2
Also known as
{{3,6}6,{6,16|2}}. if this polytope has another name.

Special Properties

  • 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

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*(s1*s0*s2)^2*s1> of order 3

16 facets

  • 16 of 3-fold non-regular quotient of {3,6}*108

5 vertex figures

Representations

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

References

None.

to this polytope.