Part of the Atlas of Small Regular Polytopes

Polytope of Type {18,4,6}

Atlas Canonical Name {18,4,6}*1728a

Overview

Group
SmallGroup(1728,46114)
Rank
4
Schläfli Type
{18,4,6}
Vertices, edges, …
18, 72, 24, 12
Order of s0s1s2s3
18
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

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

16-fold

18-fold

24-fold

36-fold

48-fold

72-fold

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

8 facets

18 vertex figures

  • 18 of 2-fold non-regular quotient of {4,6}*96

Representations

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

References

None.

to this polytope.