Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,81,4}

Atlas Canonical Name {2,81,4}*1296

Overview

Group
SmallGroup(1296,630)
Rank
4
Schläfli Type
{2,81,4}
Vertices, edges, …
2, 81, 162, 4
Order of s0s1s2s3
162
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Non-Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 39, 99)( 40,101)( 41,100)( 42,102)( 43,107)( 44,109)( 45,108)( 46,110)( 47,103)( 48,105)( 49,104)( 50,106)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 75)( 64, 77)( 65, 76)( 66, 78)( 67, 83)( 68, 85)( 69, 84)( 70, 86)( 71, 79)( 72, 81)( 73, 80)( 74, 82)(111,315)(112,317)(113,316)(114,318)(115,323)(116,325)(117,324)(118,326)(119,319)(120,321)(121,320)(122,322)(123,303)(124,305)(125,304)(126,306)(127,311)(128,313)(129,312)(130,314)(131,307)(132,309)(133,308)(134,310)(135,291)(136,293)(137,292)(138,294)(139,299)(140,301)(141,300)(142,302)(143,295)(144,297)(145,296)(146,298)(147,279)(148,281)(149,280)(150,282)(151,287)(152,289)(153,288)(154,290)(155,283)(156,285)(157,284)(158,286)(159,267)(160,269)(161,268)(162,270)(163,275)(164,277)(165,276)(166,278)(167,271)(168,273)(169,272)(170,274)(171,255)(172,257)(173,256)(174,258)(175,263)(176,265)(177,264)(178,266)(179,259)(180,261)(181,260)(182,262)(183,243)(184,245)(185,244)(186,246)(187,251)(188,253)(189,252)(190,254)(191,247)(192,249)(193,248)(194,250)(195,231)(196,233)(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)(204,237)(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)(212,229)(213,228)(214,230)(215,223)(216,225)(217,224)(218,226);;
s2 := (  3,111)(  4,112)(  5,114)(  6,113)(  7,119)(  8,120)(  9,122)( 10,121)( 11,115)( 12,116)( 13,118)( 14,117)( 15,139)( 16,140)( 17,142)( 18,141)( 19,135)( 20,136)( 21,138)( 22,137)( 23,143)( 24,144)( 25,146)( 26,145)( 27,127)( 28,128)( 29,130)( 30,129)( 31,123)( 32,124)( 33,126)( 34,125)( 35,131)( 36,132)( 37,134)( 38,133)( 39,207)( 40,208)( 41,210)( 42,209)( 43,215)( 44,216)( 45,218)( 46,217)( 47,211)( 48,212)( 49,214)( 50,213)( 51,195)( 52,196)( 53,198)( 54,197)( 55,203)( 56,204)( 57,206)( 58,205)( 59,199)( 60,200)( 61,202)( 62,201)( 63,183)( 64,184)( 65,186)( 66,185)( 67,191)( 68,192)( 69,194)( 70,193)( 71,187)( 72,188)( 73,190)( 74,189)( 75,171)( 76,172)( 77,174)( 78,173)( 79,179)( 80,180)( 81,182)( 82,181)( 83,175)( 84,176)( 85,178)( 86,177)( 87,159)( 88,160)( 89,162)( 90,161)( 91,167)( 92,168)( 93,170)( 94,169)( 95,163)( 96,164)( 97,166)( 98,165)( 99,147)(100,148)(101,150)(102,149)(103,155)(104,156)(105,158)(106,157)(107,151)(108,152)(109,154)(110,153)(219,315)(220,316)(221,318)(222,317)(223,323)(224,324)(225,326)(226,325)(227,319)(228,320)(229,322)(230,321)(231,303)(232,304)(233,306)(234,305)(235,311)(236,312)(237,314)(238,313)(239,307)(240,308)(241,310)(242,309)(243,291)(244,292)(245,294)(246,293)(247,299)(248,300)(249,302)(250,301)(251,295)(252,296)(253,298)(254,297)(255,279)(256,280)(257,282)(258,281)(259,287)(260,288)(261,290)(262,289)(263,283)(264,284)(265,286)(266,285)(269,270)(271,275)(272,276)(273,278)(274,277);;
s3 := (  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)(208,209)(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)(224,225)(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)(240,241)(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)(256,257)(259,262)(260,261)(263,266)(264,265)(267,270)(268,269)(271,274)(272,273)(275,278)(276,277)(279,282)(280,281)(283,286)(284,285)(287,290)(288,289)(291,294)(292,293)(295,298)(296,297)(299,302)(300,301)(303,306)(304,305)(307,310)(308,309)(311,314)(312,313)(315,318)(316,317)(319,322)(320,321)(323,326)(324,325);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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*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(326)!(1,2);
s1 := Sym(326)!(  4,  5)(  7, 11)(  8, 13)(  9, 12)( 10, 14)( 15, 31)( 16, 33)( 17, 32)( 18, 34)( 19, 27)( 20, 29)( 21, 28)( 22, 30)( 23, 35)( 24, 37)( 25, 36)( 26, 38)( 39, 99)( 40,101)( 41,100)( 42,102)( 43,107)( 44,109)( 45,108)( 46,110)( 47,103)( 48,105)( 49,104)( 50,106)( 51, 87)( 52, 89)( 53, 88)( 54, 90)( 55, 95)( 56, 97)( 57, 96)( 58, 98)( 59, 91)( 60, 93)( 61, 92)( 62, 94)( 63, 75)( 64, 77)( 65, 76)( 66, 78)( 67, 83)( 68, 85)( 69, 84)( 70, 86)( 71, 79)( 72, 81)( 73, 80)( 74, 82)(111,315)(112,317)(113,316)(114,318)(115,323)(116,325)(117,324)(118,326)(119,319)(120,321)(121,320)(122,322)(123,303)(124,305)(125,304)(126,306)(127,311)(128,313)(129,312)(130,314)(131,307)(132,309)(133,308)(134,310)(135,291)(136,293)(137,292)(138,294)(139,299)(140,301)(141,300)(142,302)(143,295)(144,297)(145,296)(146,298)(147,279)(148,281)(149,280)(150,282)(151,287)(152,289)(153,288)(154,290)(155,283)(156,285)(157,284)(158,286)(159,267)(160,269)(161,268)(162,270)(163,275)(164,277)(165,276)(166,278)(167,271)(168,273)(169,272)(170,274)(171,255)(172,257)(173,256)(174,258)(175,263)(176,265)(177,264)(178,266)(179,259)(180,261)(181,260)(182,262)(183,243)(184,245)(185,244)(186,246)(187,251)(188,253)(189,252)(190,254)(191,247)(192,249)(193,248)(194,250)(195,231)(196,233)(197,232)(198,234)(199,239)(200,241)(201,240)(202,242)(203,235)(204,237)(205,236)(206,238)(207,219)(208,221)(209,220)(210,222)(211,227)(212,229)(213,228)(214,230)(215,223)(216,225)(217,224)(218,226);
s2 := Sym(326)!(  3,111)(  4,112)(  5,114)(  6,113)(  7,119)(  8,120)(  9,122)( 10,121)( 11,115)( 12,116)( 13,118)( 14,117)( 15,139)( 16,140)( 17,142)( 18,141)( 19,135)( 20,136)( 21,138)( 22,137)( 23,143)( 24,144)( 25,146)( 26,145)( 27,127)( 28,128)( 29,130)( 30,129)( 31,123)( 32,124)( 33,126)( 34,125)( 35,131)( 36,132)( 37,134)( 38,133)( 39,207)( 40,208)( 41,210)( 42,209)( 43,215)( 44,216)( 45,218)( 46,217)( 47,211)( 48,212)( 49,214)( 50,213)( 51,195)( 52,196)( 53,198)( 54,197)( 55,203)( 56,204)( 57,206)( 58,205)( 59,199)( 60,200)( 61,202)( 62,201)( 63,183)( 64,184)( 65,186)( 66,185)( 67,191)( 68,192)( 69,194)( 70,193)( 71,187)( 72,188)( 73,190)( 74,189)( 75,171)( 76,172)( 77,174)( 78,173)( 79,179)( 80,180)( 81,182)( 82,181)( 83,175)( 84,176)( 85,178)( 86,177)( 87,159)( 88,160)( 89,162)( 90,161)( 91,167)( 92,168)( 93,170)( 94,169)( 95,163)( 96,164)( 97,166)( 98,165)( 99,147)(100,148)(101,150)(102,149)(103,155)(104,156)(105,158)(106,157)(107,151)(108,152)(109,154)(110,153)(219,315)(220,316)(221,318)(222,317)(223,323)(224,324)(225,326)(226,325)(227,319)(228,320)(229,322)(230,321)(231,303)(232,304)(233,306)(234,305)(235,311)(236,312)(237,314)(238,313)(239,307)(240,308)(241,310)(242,309)(243,291)(244,292)(245,294)(246,293)(247,299)(248,300)(249,302)(250,301)(251,295)(252,296)(253,298)(254,297)(255,279)(256,280)(257,282)(258,281)(259,287)(260,288)(261,290)(262,289)(263,283)(264,284)(265,286)(266,285)(269,270)(271,275)(272,276)(273,278)(274,277);
s3 := Sym(326)!(  3,  6)(  4,  5)(  7, 10)(  8,  9)( 11, 14)( 12, 13)( 15, 18)( 16, 17)( 19, 22)( 20, 21)( 23, 26)( 24, 25)( 27, 30)( 28, 29)( 31, 34)( 32, 33)( 35, 38)( 36, 37)( 39, 42)( 40, 41)( 43, 46)( 44, 45)( 47, 50)( 48, 49)( 51, 54)( 52, 53)( 55, 58)( 56, 57)( 59, 62)( 60, 61)( 63, 66)( 64, 65)( 67, 70)( 68, 69)( 71, 74)( 72, 73)( 75, 78)( 76, 77)( 79, 82)( 80, 81)( 83, 86)( 84, 85)( 87, 90)( 88, 89)( 91, 94)( 92, 93)( 95, 98)( 96, 97)( 99,102)(100,101)(103,106)(104,105)(107,110)(108,109)(111,114)(112,113)(115,118)(116,117)(119,122)(120,121)(123,126)(124,125)(127,130)(128,129)(131,134)(132,133)(135,138)(136,137)(139,142)(140,141)(143,146)(144,145)(147,150)(148,149)(151,154)(152,153)(155,158)(156,157)(159,162)(160,161)(163,166)(164,165)(167,170)(168,169)(171,174)(172,173)(175,178)(176,177)(179,182)(180,181)(183,186)(184,185)(187,190)(188,189)(191,194)(192,193)(195,198)(196,197)(199,202)(200,201)(203,206)(204,205)(207,210)(208,209)(211,214)(212,213)(215,218)(216,217)(219,222)(220,221)(223,226)(224,225)(227,230)(228,229)(231,234)(232,233)(235,238)(236,237)(239,242)(240,241)(243,246)(244,245)(247,250)(248,249)(251,254)(252,253)(255,258)(256,257)(259,262)(260,261)(263,266)(264,265)(267,270)(268,269)(271,274)(272,273)(275,278)(276,277)(279,282)(280,281)(283,286)(284,285)(287,290)(288,289)(291,294)(292,293)(295,298)(296,297)(299,302)(300,301)(303,306)(304,305)(307,310)(308,309)(311,314)(312,313)(315,318)(316,317)(319,322)(320,321)(323,326)(324,325);
poly := sub<Sym(326)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*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*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 >;