Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,27}

Atlas Canonical Name {2,6,27}*1944c

Overview

Group
SmallGroup(1944,954)
Rank
4
Schläfli Type
{2,6,27}
Vertices, edges, …
2, 18, 243, 81
Order of s0s1s2s3
54
Order of s0s1s2s3s2s1
2
Also known as
if this polytope has a name.

Special Properties

  • Degenerate
  • Universal
  • Orientable
  • Flat

Quotients maximal quotients in bold

3-fold

9-fold

27-fold

81-fold

Covers minimal covers in bold

None in this atlas.

Representations

Permutation Representation (GAP) 
s0 := (1,2);;
s1 := (  6, 11)(  7,  9)(  8, 10)( 15, 20)( 16, 18)( 17, 19)( 24, 29)( 25, 27)( 26, 28)( 30, 57)( 31, 58)( 32, 59)( 33, 65)( 34, 63)( 35, 64)( 36, 61)( 37, 62)( 38, 60)( 39, 66)( 40, 67)( 41, 68)( 42, 74)( 43, 72)( 44, 73)( 45, 70)( 46, 71)( 47, 69)( 48, 75)( 49, 76)( 50, 77)( 51, 83)( 52, 81)( 53, 82)( 54, 79)( 55, 80)( 56, 78)( 87, 92)( 88, 90)( 89, 91)( 96,101)( 97, 99)( 98,100)(105,110)(106,108)(107,109)(111,138)(112,139)(113,140)(114,146)(115,144)(116,145)(117,142)(118,143)(119,141)(120,147)(121,148)(122,149)(123,155)(124,153)(125,154)(126,151)(127,152)(128,150)(129,156)(130,157)(131,158)(132,164)(133,162)(134,163)(135,160)(136,161)(137,159)(168,173)(169,171)(170,172)(177,182)(178,180)(179,181)(186,191)(187,189)(188,190)(192,219)(193,220)(194,221)(195,227)(196,225)(197,226)(198,223)(199,224)(200,222)(201,228)(202,229)(203,230)(204,236)(205,234)(206,235)(207,232)(208,233)(209,231)(210,237)(211,238)(212,239)(213,245)(214,243)(215,244)(216,241)(217,242)(218,240);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)( 10, 38)( 11, 37)( 12, 50)( 13, 49)( 14, 48)( 15, 53)( 16, 52)( 17, 51)( 18, 56)( 19, 55)( 20, 54)( 21, 41)( 22, 40)( 23, 39)( 24, 44)( 25, 43)( 26, 42)( 27, 47)( 28, 46)( 29, 45)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 84,212)( 85,211)( 86,210)( 87,215)( 88,214)( 89,213)( 90,218)( 91,217)( 92,216)( 93,203)( 94,202)( 95,201)( 96,206)( 97,205)( 98,204)( 99,209)(100,208)(101,207)(102,194)(103,193)(104,192)(105,197)(106,196)(107,195)(108,200)(109,199)(110,198)(111,185)(112,184)(113,183)(114,188)(115,187)(116,186)(117,191)(118,190)(119,189)(120,176)(121,175)(122,174)(123,179)(124,178)(125,177)(126,182)(127,181)(128,180)(129,167)(130,166)(131,165)(132,170)(133,169)(134,168)(135,173)(136,172)(137,171)(138,239)(139,238)(140,237)(141,242)(142,241)(143,240)(144,245)(145,244)(146,243)(147,230)(148,229)(149,228)(150,233)(151,232)(152,231)(153,236)(154,235)(155,234)(156,221)(157,220)(158,219)(159,224)(160,223)(161,222)(162,227)(163,226)(164,225);;
s3 := (  3, 84)(  4, 86)(  5, 85)(  6, 88)(  7, 87)(  8, 89)(  9, 92)( 10, 91)( 11, 90)( 12,104)( 13,103)( 14,102)( 15,105)( 16,107)( 17,106)( 18,109)( 19,108)( 20,110)( 21, 95)( 22, 94)( 23, 93)( 24, 96)( 25, 98)( 26, 97)( 27,100)( 28, 99)( 29,101)( 30,146)( 31,145)( 32,144)( 33,138)( 34,140)( 35,139)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)( 42,158)( 43,157)( 44,156)( 45,159)( 46,161)( 47,160)( 48,154)( 49,153)( 50,155)( 51,149)( 52,148)( 53,147)( 54,150)( 55,152)( 56,151)( 57,114)( 58,116)( 59,115)( 60,118)( 61,117)( 62,119)( 63,113)( 64,112)( 65,111)( 66,134)( 67,133)( 68,132)( 69,135)( 70,137)( 71,136)( 72,130)( 73,129)( 74,131)( 75,125)( 76,124)( 77,123)( 78,126)( 79,128)( 80,127)( 81,121)( 82,120)( 83,122)(165,185)(166,184)(167,183)(168,186)(169,188)(170,187)(171,190)(172,189)(173,191)(174,176)(178,179)(180,181)(192,244)(193,243)(194,245)(195,239)(196,238)(197,237)(198,240)(199,242)(200,241)(201,235)(202,234)(203,236)(204,230)(205,229)(206,228)(207,231)(208,233)(209,232)(210,226)(211,225)(212,227)(213,221)(214,220)(215,219)(216,222)(217,224)(218,223);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s3*s2*s3*s2*s1*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  6, 11)(  7,  9)(  8, 10)( 15, 20)( 16, 18)( 17, 19)( 24, 29)( 25, 27)( 26, 28)( 30, 57)( 31, 58)( 32, 59)( 33, 65)( 34, 63)( 35, 64)( 36, 61)( 37, 62)( 38, 60)( 39, 66)( 40, 67)( 41, 68)( 42, 74)( 43, 72)( 44, 73)( 45, 70)( 46, 71)( 47, 69)( 48, 75)( 49, 76)( 50, 77)( 51, 83)( 52, 81)( 53, 82)( 54, 79)( 55, 80)( 56, 78)( 87, 92)( 88, 90)( 89, 91)( 96,101)( 97, 99)( 98,100)(105,110)(106,108)(107,109)(111,138)(112,139)(113,140)(114,146)(115,144)(116,145)(117,142)(118,143)(119,141)(120,147)(121,148)(122,149)(123,155)(124,153)(125,154)(126,151)(127,152)(128,150)(129,156)(130,157)(131,158)(132,164)(133,162)(134,163)(135,160)(136,161)(137,159)(168,173)(169,171)(170,172)(177,182)(178,180)(179,181)(186,191)(187,189)(188,190)(192,219)(193,220)(194,221)(195,227)(196,225)(197,226)(198,223)(199,224)(200,222)(201,228)(202,229)(203,230)(204,236)(205,234)(206,235)(207,232)(208,233)(209,231)(210,237)(211,238)(212,239)(213,245)(214,243)(215,244)(216,241)(217,242)(218,240);
s2 := Sym(245)!(  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)( 10, 38)( 11, 37)( 12, 50)( 13, 49)( 14, 48)( 15, 53)( 16, 52)( 17, 51)( 18, 56)( 19, 55)( 20, 54)( 21, 41)( 22, 40)( 23, 39)( 24, 44)( 25, 43)( 26, 42)( 27, 47)( 28, 46)( 29, 45)( 58, 59)( 61, 62)( 64, 65)( 66, 77)( 67, 76)( 68, 75)( 69, 80)( 70, 79)( 71, 78)( 72, 83)( 73, 82)( 74, 81)( 84,212)( 85,211)( 86,210)( 87,215)( 88,214)( 89,213)( 90,218)( 91,217)( 92,216)( 93,203)( 94,202)( 95,201)( 96,206)( 97,205)( 98,204)( 99,209)(100,208)(101,207)(102,194)(103,193)(104,192)(105,197)(106,196)(107,195)(108,200)(109,199)(110,198)(111,185)(112,184)(113,183)(114,188)(115,187)(116,186)(117,191)(118,190)(119,189)(120,176)(121,175)(122,174)(123,179)(124,178)(125,177)(126,182)(127,181)(128,180)(129,167)(130,166)(131,165)(132,170)(133,169)(134,168)(135,173)(136,172)(137,171)(138,239)(139,238)(140,237)(141,242)(142,241)(143,240)(144,245)(145,244)(146,243)(147,230)(148,229)(149,228)(150,233)(151,232)(152,231)(153,236)(154,235)(155,234)(156,221)(157,220)(158,219)(159,224)(160,223)(161,222)(162,227)(163,226)(164,225);
s3 := Sym(245)!(  3, 84)(  4, 86)(  5, 85)(  6, 88)(  7, 87)(  8, 89)(  9, 92)( 10, 91)( 11, 90)( 12,104)( 13,103)( 14,102)( 15,105)( 16,107)( 17,106)( 18,109)( 19,108)( 20,110)( 21, 95)( 22, 94)( 23, 93)( 24, 96)( 25, 98)( 26, 97)( 27,100)( 28, 99)( 29,101)( 30,146)( 31,145)( 32,144)( 33,138)( 34,140)( 35,139)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)( 42,158)( 43,157)( 44,156)( 45,159)( 46,161)( 47,160)( 48,154)( 49,153)( 50,155)( 51,149)( 52,148)( 53,147)( 54,150)( 55,152)( 56,151)( 57,114)( 58,116)( 59,115)( 60,118)( 61,117)( 62,119)( 63,113)( 64,112)( 65,111)( 66,134)( 67,133)( 68,132)( 69,135)( 70,137)( 71,136)( 72,130)( 73,129)( 74,131)( 75,125)( 76,124)( 77,123)( 78,126)( 79,128)( 80,127)( 81,121)( 82,120)( 83,122)(165,185)(166,184)(167,183)(168,186)(169,188)(170,187)(171,190)(172,189)(173,191)(174,176)(178,179)(180,181)(192,244)(193,243)(194,245)(195,239)(196,238)(197,237)(198,240)(199,242)(200,241)(201,235)(202,234)(203,236)(204,230)(205,229)(206,228)(207,231)(208,233)(209,232)(210,226)(211,225)(212,227)(213,221)(214,220)(215,219)(216,222)(217,224)(218,223);
poly := sub<Sym(245)|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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s3*s2*s3*s2*s1*s3*s2*s3*s2*s3 >;