Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,9,18}

Atlas Canonical Name {2,9,18}*1944a

Overview

Group
SmallGroup(1944,940)
Rank
4
Schläfli Type
{2,9,18}
Vertices, edges, …
2, 27, 243, 54
Order of s0s1s2s3
18
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,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 30, 63)( 31, 64)( 32, 65)( 33, 60)( 34, 61)( 35, 62)( 36, 57)( 37, 58)( 38, 59)( 39, 81)( 40, 82)( 41, 83)( 42, 78)( 43, 79)( 44, 80)( 45, 75)( 46, 76)( 47, 77)( 48, 72)( 49, 73)( 50, 74)( 51, 69)( 52, 70)( 53, 71)( 54, 66)( 55, 67)( 56, 68)( 84,183)( 85,184)( 86,185)( 87,189)( 88,190)( 89,191)( 90,186)( 91,187)( 92,188)( 93,174)( 94,175)( 95,176)( 96,180)( 97,181)( 98,182)( 99,177)(100,178)(101,179)(102,165)(103,166)(104,167)(105,171)(106,172)(107,173)(108,168)(109,169)(110,170)(111,243)(112,244)(113,245)(114,240)(115,241)(116,242)(117,237)(118,238)(119,239)(120,234)(121,235)(122,236)(123,231)(124,232)(125,233)(126,228)(127,229)(128,230)(129,225)(130,226)(131,227)(132,222)(133,223)(134,224)(135,219)(136,220)(137,221)(138,216)(139,217)(140,218)(141,213)(142,214)(143,215)(144,210)(145,211)(146,212)(147,207)(148,208)(149,209)(150,204)(151,205)(152,206)(153,201)(154,202)(155,203)(156,198)(157,199)(158,200)(159,195)(160,196)(161,197)(162,192)(163,193)(164,194);;
s2 := (  3,193)(  4,194)(  5,192)(  6,199)(  7,200)(  8,198)(  9,196)( 10,197)( 11,195)( 12,211)( 13,212)( 14,210)( 15,217)( 16,218)( 17,216)( 18,214)( 19,215)( 20,213)( 21,202)( 22,203)( 23,201)( 24,208)( 25,209)( 26,207)( 27,205)( 28,206)( 29,204)( 30,167)( 31,165)( 32,166)( 33,173)( 34,171)( 35,172)( 36,170)( 37,168)( 38,169)( 39,185)( 40,183)( 41,184)( 42,191)( 43,189)( 44,190)( 45,188)( 46,186)( 47,187)( 48,176)( 49,174)( 50,175)( 51,182)( 52,180)( 53,181)( 54,179)( 55,177)( 56,178)( 57,225)( 58,226)( 59,227)( 60,222)( 61,223)( 62,224)( 63,219)( 64,220)( 65,221)( 66,243)( 67,244)( 68,245)( 69,240)( 70,241)( 71,242)( 72,237)( 73,238)( 74,239)( 75,234)( 76,235)( 77,236)( 78,231)( 79,232)( 80,233)( 81,228)( 82,229)( 83,230)( 84,112)( 85,113)( 86,111)( 87,118)( 88,119)( 89,117)( 90,115)( 91,116)( 92,114)( 93,130)( 94,131)( 95,129)( 96,136)( 97,137)( 98,135)( 99,133)(100,134)(101,132)(102,121)(103,122)(104,120)(105,127)(106,128)(107,126)(108,124)(109,125)(110,123)(138,144)(139,145)(140,146)(147,162)(148,163)(149,164)(150,159)(151,160)(152,161)(153,156)(154,157)(155,158);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)(197,223)(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)(205,233)(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)(213,240)(214,242)(215,241)(216,237)(217,239)(218,238);;
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*s3*s2*s3*s1*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*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(245)!(1,2);
s1 := Sym(245)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 30, 63)( 31, 64)( 32, 65)( 33, 60)( 34, 61)( 35, 62)( 36, 57)( 37, 58)( 38, 59)( 39, 81)( 40, 82)( 41, 83)( 42, 78)( 43, 79)( 44, 80)( 45, 75)( 46, 76)( 47, 77)( 48, 72)( 49, 73)( 50, 74)( 51, 69)( 52, 70)( 53, 71)( 54, 66)( 55, 67)( 56, 68)( 84,183)( 85,184)( 86,185)( 87,189)( 88,190)( 89,191)( 90,186)( 91,187)( 92,188)( 93,174)( 94,175)( 95,176)( 96,180)( 97,181)( 98,182)( 99,177)(100,178)(101,179)(102,165)(103,166)(104,167)(105,171)(106,172)(107,173)(108,168)(109,169)(110,170)(111,243)(112,244)(113,245)(114,240)(115,241)(116,242)(117,237)(118,238)(119,239)(120,234)(121,235)(122,236)(123,231)(124,232)(125,233)(126,228)(127,229)(128,230)(129,225)(130,226)(131,227)(132,222)(133,223)(134,224)(135,219)(136,220)(137,221)(138,216)(139,217)(140,218)(141,213)(142,214)(143,215)(144,210)(145,211)(146,212)(147,207)(148,208)(149,209)(150,204)(151,205)(152,206)(153,201)(154,202)(155,203)(156,198)(157,199)(158,200)(159,195)(160,196)(161,197)(162,192)(163,193)(164,194);
s2 := Sym(245)!(  3,193)(  4,194)(  5,192)(  6,199)(  7,200)(  8,198)(  9,196)( 10,197)( 11,195)( 12,211)( 13,212)( 14,210)( 15,217)( 16,218)( 17,216)( 18,214)( 19,215)( 20,213)( 21,202)( 22,203)( 23,201)( 24,208)( 25,209)( 26,207)( 27,205)( 28,206)( 29,204)( 30,167)( 31,165)( 32,166)( 33,173)( 34,171)( 35,172)( 36,170)( 37,168)( 38,169)( 39,185)( 40,183)( 41,184)( 42,191)( 43,189)( 44,190)( 45,188)( 46,186)( 47,187)( 48,176)( 49,174)( 50,175)( 51,182)( 52,180)( 53,181)( 54,179)( 55,177)( 56,178)( 57,225)( 58,226)( 59,227)( 60,222)( 61,223)( 62,224)( 63,219)( 64,220)( 65,221)( 66,243)( 67,244)( 68,245)( 69,240)( 70,241)( 71,242)( 72,237)( 73,238)( 74,239)( 75,234)( 76,235)( 77,236)( 78,231)( 79,232)( 80,233)( 81,228)( 82,229)( 83,230)( 84,112)( 85,113)( 86,111)( 87,118)( 88,119)( 89,117)( 90,115)( 91,116)( 92,114)( 93,130)( 94,131)( 95,129)( 96,136)( 97,137)( 98,135)( 99,133)(100,134)(101,132)(102,121)(103,122)(104,120)(105,127)(106,128)(107,126)(108,124)(109,125)(110,123)(138,144)(139,145)(140,146)(147,162)(148,163)(149,164)(150,159)(151,160)(152,161)(153,156)(154,157)(155,158);
s3 := Sym(245)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)(197,223)(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)(205,233)(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)(213,240)(214,242)(215,241)(216,237)(217,239)(218,238);
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*s3*s2*s3*s1*s2*s1*s2*s1*s2*s3*s2*s3*s1*s2, 
s1*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;