Polytope of Type {4,4,27} with group SmallGroup(1728,11358)

Overview

Group: SmallGroup(1728,11358)
Rank: 4
Schläfli Type: {4,4,27}
Vertices, edges, …: 4, 16, 108, 54
Order of s₀s₁s₂s₃: 108
Order of s₀s₁s₂s₃s₂s₁: 2
Also known as: if this polytope has a name.

Special Properties

Universal
Orientable
Flat

Quotients maximal quotients in bold

2-fold

{2,4,27}*864

3-fold

{4,4,9}*576b

4-fold

6-fold

{2,4,9}*288

8-fold

{2,2,27}*216

9-fold

{4,4,3}*192b

12-fold

18-fold

{2,4,3}*96

24-fold

{2,2,9}*72

36-fold

72-fold

{2,2,3}*24

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=<(s1*s2)^2> of order 2

36 facets

18 of {4,2}*16
18 of {4,4}*32

4 vertex figures

4 of 2-fold non-regular quotient of {4,27}*432

Representations

Permutation Representation (GAP)

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