Part of the Atlas of Small Regular Polytopes

Polytope of Type {3,18,4}

Atlas Canonical Name {3,18,4}*1296

Overview

Group
SmallGroup(1296,943)
Rank
4
Schläfli Type
{3,18,4}
Vertices, edges, …
9, 81, 108, 4
Order of s0s1s2s3
12
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

6-fold

9-fold

18-fold

27-fold

54-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*s1)^2*s0*s2*s1*s0*s2> of order 3

4 facets

3 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)( 31, 59)( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)(113,139)(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)(121,158)(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)(129,145)(130,149)(131,148)(132,150)(133,151)(134,153)(135,152)(164,165)(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)(179,187)(180,189)(190,219)(191,218)(192,217)(193,221)(194,220)(195,222)(196,223)(197,225)(198,224)(199,237)(200,236)(201,235)(202,239)(203,238)(204,240)(205,241)(206,243)(207,242)(208,228)(209,227)(210,226)(211,230)(212,229)(213,231)(214,232)(215,234)(216,233)(245,246)(247,249)(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)(261,270)(271,300)(272,299)(273,298)(274,302)(275,301)(276,303)(277,304)(278,306)(279,305)(280,318)(281,317)(282,316)(283,320)(284,319)(285,321)(286,322)(287,324)(288,323)(289,309)(290,308)(291,307)(292,311)(293,310)(294,312)(295,313)(296,315)(297,314);;
s1 := (  1, 67)(  2, 69)(  3, 68)(  4, 70)(  5, 72)(  6, 71)(  7, 64)(  8, 66)(  9, 65)( 10, 61)( 11, 63)( 12, 62)( 13, 55)( 14, 57)( 15, 56)( 16, 58)( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 78)( 23, 77)( 24, 76)( 25, 81)( 26, 80)( 27, 79)( 28, 40)( 29, 42)( 30, 41)( 31, 43)( 32, 45)( 33, 44)( 34, 37)( 35, 39)( 36, 38)( 46, 48)( 49, 51)( 52, 54)( 82,148)( 83,150)( 84,149)( 85,151)( 86,153)( 87,152)( 88,145)( 89,147)( 90,146)( 91,142)( 92,144)( 93,143)( 94,136)( 95,138)( 96,137)( 97,139)( 98,141)( 99,140)(100,156)(101,155)(102,154)(103,159)(104,158)(105,157)(106,162)(107,161)(108,160)(109,121)(110,123)(111,122)(112,124)(113,126)(114,125)(115,118)(116,120)(117,119)(127,129)(130,132)(133,135)(163,229)(164,231)(165,230)(166,232)(167,234)(168,233)(169,226)(170,228)(171,227)(172,223)(173,225)(174,224)(175,217)(176,219)(177,218)(178,220)(179,222)(180,221)(181,237)(182,236)(183,235)(184,240)(185,239)(186,238)(187,243)(188,242)(189,241)(190,202)(191,204)(192,203)(193,205)(194,207)(195,206)(196,199)(197,201)(198,200)(208,210)(211,213)(214,216)(244,310)(245,312)(246,311)(247,313)(248,315)(249,314)(250,307)(251,309)(252,308)(253,304)(254,306)(255,305)(256,298)(257,300)(258,299)(259,301)(260,303)(261,302)(262,318)(263,317)(264,316)(265,321)(266,320)(267,319)(268,324)(269,323)(270,322)(271,283)(272,285)(273,284)(274,286)(275,288)(276,287)(277,280)(278,282)(279,281)(289,291)(292,294)(295,297);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 57)( 29, 56)( 30, 55)( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)( 47, 74)( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,138)(110,137)(111,136)(112,144)(113,143)(114,142)(115,141)(116,140)(117,139)(118,147)(119,146)(120,145)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,156)(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,300)(191,299)(192,298)(193,306)(194,305)(195,304)(196,303)(197,302)(198,301)(199,309)(200,308)(201,307)(202,315)(203,314)(204,313)(205,312)(206,311)(207,310)(208,318)(209,317)(210,316)(211,324)(212,323)(213,322)(214,321)(215,320)(216,319)(217,273)(218,272)(219,271)(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,282)(227,281)(228,280)(229,288)(230,287)(231,286)(232,285)(233,284)(234,283)(235,291)(236,290)(237,289)(238,297)(239,296)(240,295)(241,294)(242,293)(243,292);;
s3 := (  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)(  8,170)(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324);;
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(324)!(  2,  3)(  4,  6)(  7,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28, 57)( 29, 56)( 30, 55)( 31, 59)( 32, 58)( 33, 60)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 83, 84)( 85, 87)( 88, 89)( 91,100)( 92,102)( 93,101)( 94,105)( 95,104)( 96,103)( 97,107)( 98,106)( 99,108)(109,138)(110,137)(111,136)(112,140)(113,139)(114,141)(115,142)(116,144)(117,143)(118,156)(119,155)(120,154)(121,158)(122,157)(123,159)(124,160)(125,162)(126,161)(127,147)(128,146)(129,145)(130,149)(131,148)(132,150)(133,151)(134,153)(135,152)(164,165)(166,168)(169,170)(172,181)(173,183)(174,182)(175,186)(176,185)(177,184)(178,188)(179,187)(180,189)(190,219)(191,218)(192,217)(193,221)(194,220)(195,222)(196,223)(197,225)(198,224)(199,237)(200,236)(201,235)(202,239)(203,238)(204,240)(205,241)(206,243)(207,242)(208,228)(209,227)(210,226)(211,230)(212,229)(213,231)(214,232)(215,234)(216,233)(245,246)(247,249)(250,251)(253,262)(254,264)(255,263)(256,267)(257,266)(258,265)(259,269)(260,268)(261,270)(271,300)(272,299)(273,298)(274,302)(275,301)(276,303)(277,304)(278,306)(279,305)(280,318)(281,317)(282,316)(283,320)(284,319)(285,321)(286,322)(287,324)(288,323)(289,309)(290,308)(291,307)(292,311)(293,310)(294,312)(295,313)(296,315)(297,314);
s1 := Sym(324)!(  1, 67)(  2, 69)(  3, 68)(  4, 70)(  5, 72)(  6, 71)(  7, 64)(  8, 66)(  9, 65)( 10, 61)( 11, 63)( 12, 62)( 13, 55)( 14, 57)( 15, 56)( 16, 58)( 17, 60)( 18, 59)( 19, 75)( 20, 74)( 21, 73)( 22, 78)( 23, 77)( 24, 76)( 25, 81)( 26, 80)( 27, 79)( 28, 40)( 29, 42)( 30, 41)( 31, 43)( 32, 45)( 33, 44)( 34, 37)( 35, 39)( 36, 38)( 46, 48)( 49, 51)( 52, 54)( 82,148)( 83,150)( 84,149)( 85,151)( 86,153)( 87,152)( 88,145)( 89,147)( 90,146)( 91,142)( 92,144)( 93,143)( 94,136)( 95,138)( 96,137)( 97,139)( 98,141)( 99,140)(100,156)(101,155)(102,154)(103,159)(104,158)(105,157)(106,162)(107,161)(108,160)(109,121)(110,123)(111,122)(112,124)(113,126)(114,125)(115,118)(116,120)(117,119)(127,129)(130,132)(133,135)(163,229)(164,231)(165,230)(166,232)(167,234)(168,233)(169,226)(170,228)(171,227)(172,223)(173,225)(174,224)(175,217)(176,219)(177,218)(178,220)(179,222)(180,221)(181,237)(182,236)(183,235)(184,240)(185,239)(186,238)(187,243)(188,242)(189,241)(190,202)(191,204)(192,203)(193,205)(194,207)(195,206)(196,199)(197,201)(198,200)(208,210)(211,213)(214,216)(244,310)(245,312)(246,311)(247,313)(248,315)(249,314)(250,307)(251,309)(252,308)(253,304)(254,306)(255,305)(256,298)(257,300)(258,299)(259,301)(260,303)(261,302)(262,318)(263,317)(264,316)(265,321)(266,320)(267,319)(268,324)(269,323)(270,322)(271,283)(272,285)(273,284)(274,286)(275,288)(276,287)(277,280)(278,282)(279,281)(289,291)(292,294)(295,297);
s2 := Sym(324)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 57)( 29, 56)( 30, 55)( 31, 63)( 32, 62)( 33, 61)( 34, 60)( 35, 59)( 36, 58)( 37, 66)( 38, 65)( 39, 64)( 40, 72)( 41, 71)( 42, 70)( 43, 69)( 44, 68)( 45, 67)( 46, 75)( 47, 74)( 48, 73)( 49, 81)( 50, 80)( 51, 79)( 52, 78)( 53, 77)( 54, 76)( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)(109,138)(110,137)(111,136)(112,144)(113,143)(114,142)(115,141)(116,140)(117,139)(118,147)(119,146)(120,145)(121,153)(122,152)(123,151)(124,150)(125,149)(126,148)(127,156)(128,155)(129,154)(130,162)(131,161)(132,160)(133,159)(134,158)(135,157)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)(187,265)(188,267)(189,266)(190,300)(191,299)(192,298)(193,306)(194,305)(195,304)(196,303)(197,302)(198,301)(199,309)(200,308)(201,307)(202,315)(203,314)(204,313)(205,312)(206,311)(207,310)(208,318)(209,317)(210,316)(211,324)(212,323)(213,322)(214,321)(215,320)(216,319)(217,273)(218,272)(219,271)(220,279)(221,278)(222,277)(223,276)(224,275)(225,274)(226,282)(227,281)(228,280)(229,288)(230,287)(231,286)(232,285)(233,284)(234,283)(235,291)(236,290)(237,289)(238,297)(239,296)(240,295)(241,294)(242,293)(243,292);
s3 := Sym(324)!(  1,163)(  2,164)(  3,165)(  4,166)(  5,167)(  6,168)(  7,169)(  8,170)(  9,171)( 10,172)( 11,173)( 12,174)( 13,175)( 14,176)( 15,177)( 16,178)( 17,179)( 18,180)( 19,181)( 20,182)( 21,183)( 22,184)( 23,185)( 24,186)( 25,187)( 26,188)( 27,189)( 28,190)( 29,191)( 30,192)( 31,193)( 32,194)( 33,195)( 34,196)( 35,197)( 36,198)( 37,199)( 38,200)( 39,201)( 40,202)( 41,203)( 42,204)( 43,205)( 44,206)( 45,207)( 46,208)( 47,209)( 48,210)( 49,211)( 50,212)( 51,213)( 52,214)( 53,215)( 54,216)( 55,217)( 56,218)( 57,219)( 58,220)( 59,221)( 60,222)( 61,223)( 62,224)( 63,225)( 64,226)( 65,227)( 66,228)( 67,229)( 68,230)( 69,231)( 70,232)( 71,233)( 72,234)( 73,235)( 74,236)( 75,237)( 76,238)( 77,239)( 78,240)( 79,241)( 80,242)( 81,243)( 82,244)( 83,245)( 84,246)( 85,247)( 86,248)( 87,249)( 88,250)( 89,251)( 90,252)( 91,253)( 92,254)( 93,255)( 94,256)( 95,257)( 96,258)( 97,259)( 98,260)( 99,261)(100,262)(101,263)(102,264)(103,265)(104,266)(105,267)(106,268)(107,269)(108,270)(109,271)(110,272)(111,273)(112,274)(113,275)(114,276)(115,277)(116,278)(117,279)(118,280)(119,281)(120,282)(121,283)(122,284)(123,285)(124,286)(125,287)(126,288)(127,289)(128,290)(129,291)(130,292)(131,293)(132,294)(133,295)(134,296)(135,297)(136,298)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,305)(144,306)(145,307)(146,308)(147,309)(148,310)(149,311)(150,312)(151,313)(152,314)(153,315)(154,316)(155,317)(156,318)(157,319)(158,320)(159,321)(160,322)(161,323)(162,324);
poly := sub<Sym(324)|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, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1 >;

References

None.

to this polytope.