Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,18}

Atlas Canonical Name {2,18,18}*1944d

Overview

Group
SmallGroup(1944,945)
Rank
4
Schläfli Type
{2,18,18}
Vertices, edges, …
2, 27, 243, 27
Order of s0s1s2s3
18
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 := (  4,  5)(  7,  8)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 27)( 16, 29)( 17, 28)( 18, 21)( 19, 23)( 20, 22)( 30, 58)( 31, 57)( 32, 59)( 33, 61)( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)( 41, 80)( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)( 49, 72)( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)(115,141)(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)(123,163)(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)(131,155)(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)(197,224)(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)(205,243)(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)(213,229)(214,228)(215,230)(216,232)(217,231)(218,233);;
s2 := (  3, 30)(  4, 32)(  5, 31)(  6, 36)(  7, 38)(  8, 37)(  9, 33)( 10, 35)( 11, 34)( 12, 41)( 13, 40)( 14, 39)( 15, 47)( 16, 46)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)( 26, 56)( 27, 52)( 28, 51)( 29, 53)( 57, 58)( 60, 64)( 61, 63)( 62, 65)( 67, 68)( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)( 91,200)( 92,199)( 93,206)( 94,205)( 95,204)( 96,203)( 97,202)( 98,201)( 99,209)(100,208)(101,207)(102,214)(103,213)(104,215)(105,211)(106,210)(107,212)(108,217)(109,216)(110,218)(111,168)(112,170)(113,169)(114,165)(115,167)(116,166)(117,171)(118,173)(119,172)(120,179)(121,178)(122,177)(123,176)(124,175)(125,174)(126,182)(127,181)(128,180)(129,187)(130,186)(131,188)(132,184)(133,183)(134,185)(135,190)(136,189)(137,191)(138,223)(139,222)(140,224)(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)(148,233)(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)(156,242)(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)(164,243);;
s3 := (  3, 84)(  4, 85)(  5, 86)(  6, 90)(  7, 91)(  8, 92)(  9, 87)( 10, 88)( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)( 18,105)( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)( 26,101)( 27, 96)( 28, 97)( 29, 98)( 30,120)( 31,121)( 32,122)( 33,126)( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)( 42,117)( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)( 50,131)( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,159)( 58,160)( 59,161)( 60,156)( 61,157)( 62,158)( 63,162)( 64,163)( 65,164)( 66,150)( 67,151)( 68,152)( 69,147)( 70,148)( 71,149)( 72,153)( 73,154)( 74,155)( 75,141)( 76,142)( 77,143)( 78,138)( 79,139)( 80,140)( 81,144)( 82,145)( 83,146)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)(178,184)(179,185)(180,189)(181,190)(182,191)(192,204)(193,205)(194,206)(195,201)(196,202)(197,203)(198,207)(199,208)(200,209)(210,213)(211,214)(212,215)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)(225,237)(226,238)(227,239)(228,234)(229,235)(230,236);;
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, 
s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1, 
s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  7,  8)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 27)( 16, 29)( 17, 28)( 18, 21)( 19, 23)( 20, 22)( 30, 58)( 31, 57)( 32, 59)( 33, 61)( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)( 41, 80)( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)( 49, 72)( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)(115,141)(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)(123,163)(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)(131,155)(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)(197,224)(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)(205,243)(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)(213,229)(214,228)(215,230)(216,232)(217,231)(218,233);
s2 := Sym(245)!(  3, 30)(  4, 32)(  5, 31)(  6, 36)(  7, 38)(  8, 37)(  9, 33)( 10, 35)( 11, 34)( 12, 41)( 13, 40)( 14, 39)( 15, 47)( 16, 46)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)( 26, 56)( 27, 52)( 28, 51)( 29, 53)( 57, 58)( 60, 64)( 61, 63)( 62, 65)( 67, 68)( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)( 91,200)( 92,199)( 93,206)( 94,205)( 95,204)( 96,203)( 97,202)( 98,201)( 99,209)(100,208)(101,207)(102,214)(103,213)(104,215)(105,211)(106,210)(107,212)(108,217)(109,216)(110,218)(111,168)(112,170)(113,169)(114,165)(115,167)(116,166)(117,171)(118,173)(119,172)(120,179)(121,178)(122,177)(123,176)(124,175)(125,174)(126,182)(127,181)(128,180)(129,187)(130,186)(131,188)(132,184)(133,183)(134,185)(135,190)(136,189)(137,191)(138,223)(139,222)(140,224)(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)(148,233)(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)(156,242)(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)(164,243);
s3 := Sym(245)!(  3, 84)(  4, 85)(  5, 86)(  6, 90)(  7, 91)(  8, 92)(  9, 87)( 10, 88)( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)( 18,105)( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)( 26,101)( 27, 96)( 28, 97)( 29, 98)( 30,120)( 31,121)( 32,122)( 33,126)( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)( 42,117)( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)( 50,131)( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,159)( 58,160)( 59,161)( 60,156)( 61,157)( 62,158)( 63,162)( 64,163)( 65,164)( 66,150)( 67,151)( 68,152)( 69,147)( 70,148)( 71,149)( 72,153)( 73,154)( 74,155)( 75,141)( 76,142)( 77,143)( 78,138)( 79,139)( 80,140)( 81,144)( 82,145)( 83,146)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)(178,184)(179,185)(180,189)(181,190)(182,191)(192,204)(193,205)(194,206)(195,201)(196,202)(197,203)(198,207)(199,208)(200,209)(210,213)(211,214)(212,215)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)(225,237)(226,238)(227,239)(228,234)(229,235)(230,236);
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, s3*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1, 
s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2 >;