Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,54}

Atlas Canonical Name {2,6,54}*1944a

Overview

Group
SmallGroup(1944,948)
Rank
4
Schläfli Type
{2,6,54}
Vertices, edges, …
2, 9, 243, 81
Order of s0s1s2s3
54
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

Covers minimal covers in bold

None in this atlas.

Representations

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