Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,9,18}

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

Overview

Group
SmallGroup(1944,946)
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 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 59)( 31, 58)( 32, 57)( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 73)( 40, 72)( 41, 74)( 42, 70)( 43, 69)( 44, 71)( 45, 67)( 46, 66)( 47, 68)( 48, 78)( 49, 80)( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 81)( 55, 83)( 56, 82)( 84,168)( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)( 90,171)( 91,173)( 92,172)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)(101,177)(102,190)(103,189)(104,191)(105,187)(106,186)(107,188)(108,184)(109,183)(110,185)(111,224)(112,223)(113,222)(114,221)(115,220)(116,219)(117,227)(118,226)(119,225)(120,229)(121,228)(122,230)(123,235)(124,234)(125,236)(126,232)(127,231)(128,233)(129,243)(130,245)(131,244)(132,240)(133,242)(134,241)(135,237)(136,239)(137,238)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,200)(145,199)(146,198)(147,202)(148,201)(149,203)(150,208)(151,207)(152,209)(153,205)(154,204)(155,206)(156,216)(157,218)(158,217)(159,213)(160,215)(161,214)(162,210)(163,212)(164,211);;
s2 := (  3,201)(  4,203)(  5,202)(  6,207)(  7,209)(  8,208)(  9,204)( 10,206)( 11,205)( 12,211)( 13,210)( 14,212)( 15,217)( 16,216)( 17,218)( 18,214)( 19,213)( 20,215)( 21,194)( 22,193)( 23,192)( 24,200)( 25,199)( 26,198)( 27,197)( 28,196)( 29,195)( 30,185)( 31,184)( 32,183)( 33,191)( 34,190)( 35,189)( 36,188)( 37,187)( 38,186)( 39,165)( 40,167)( 41,166)( 42,171)( 43,173)( 44,172)( 45,168)( 46,170)( 47,169)( 48,175)( 49,174)( 50,176)( 51,181)( 52,180)( 53,182)( 54,178)( 55,177)( 56,179)( 57,225)( 58,227)( 59,226)( 60,222)( 61,224)( 62,223)( 63,219)( 64,221)( 65,220)( 66,235)( 67,234)( 68,236)( 69,232)( 70,231)( 71,233)( 72,229)( 73,228)( 74,230)( 75,245)( 76,244)( 77,243)( 78,242)( 79,241)( 80,240)( 81,239)( 82,238)( 83,237)( 84,120)( 85,122)( 86,121)( 87,126)( 88,128)( 89,127)( 90,123)( 91,125)( 92,124)( 93,130)( 94,129)( 95,131)( 96,136)( 97,135)( 98,137)( 99,133)(100,132)(101,134)(102,113)(103,112)(104,111)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(138,144)(139,146)(140,145)(142,143)(147,154)(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);;
s3 := (  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, 59)( 31, 58)( 32, 57)( 33, 62)( 34, 61)( 35, 60)( 36, 65)( 37, 64)( 38, 63)( 39, 80)( 40, 79)( 41, 78)( 42, 83)( 43, 82)( 44, 81)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)( 50, 72)( 51, 68)( 52, 67)( 53, 66)( 54, 71)( 55, 70)( 56, 69)( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)( 99,102)(100,104)(101,103)(111,140)(112,139)(113,138)(114,143)(115,142)(116,141)(117,146)(118,145)(119,144)(120,161)(121,160)(122,159)(123,164)(124,163)(125,162)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,149)(133,148)(134,147)(135,152)(136,151)(137,150)(166,167)(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)(181,185)(182,184)(192,221)(193,220)(194,219)(195,224)(196,223)(197,222)(198,227)(199,226)(200,225)(201,242)(202,241)(203,240)(204,245)(205,244)(206,243)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)(213,230)(214,229)(215,228)(216,233)(217,232)(218,231);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 59)( 31, 58)( 32, 57)( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 73)( 40, 72)( 41, 74)( 42, 70)( 43, 69)( 44, 71)( 45, 67)( 46, 66)( 47, 68)( 48, 78)( 49, 80)( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 81)( 55, 83)( 56, 82)( 84,168)( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)( 90,171)( 91,173)( 92,172)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)(101,177)(102,190)(103,189)(104,191)(105,187)(106,186)(107,188)(108,184)(109,183)(110,185)(111,224)(112,223)(113,222)(114,221)(115,220)(116,219)(117,227)(118,226)(119,225)(120,229)(121,228)(122,230)(123,235)(124,234)(125,236)(126,232)(127,231)(128,233)(129,243)(130,245)(131,244)(132,240)(133,242)(134,241)(135,237)(136,239)(137,238)(138,197)(139,196)(140,195)(141,194)(142,193)(143,192)(144,200)(145,199)(146,198)(147,202)(148,201)(149,203)(150,208)(151,207)(152,209)(153,205)(154,204)(155,206)(156,216)(157,218)(158,217)(159,213)(160,215)(161,214)(162,210)(163,212)(164,211);
s2 := Sym(245)!(  3,201)(  4,203)(  5,202)(  6,207)(  7,209)(  8,208)(  9,204)( 10,206)( 11,205)( 12,211)( 13,210)( 14,212)( 15,217)( 16,216)( 17,218)( 18,214)( 19,213)( 20,215)( 21,194)( 22,193)( 23,192)( 24,200)( 25,199)( 26,198)( 27,197)( 28,196)( 29,195)( 30,185)( 31,184)( 32,183)( 33,191)( 34,190)( 35,189)( 36,188)( 37,187)( 38,186)( 39,165)( 40,167)( 41,166)( 42,171)( 43,173)( 44,172)( 45,168)( 46,170)( 47,169)( 48,175)( 49,174)( 50,176)( 51,181)( 52,180)( 53,182)( 54,178)( 55,177)( 56,179)( 57,225)( 58,227)( 59,226)( 60,222)( 61,224)( 62,223)( 63,219)( 64,221)( 65,220)( 66,235)( 67,234)( 68,236)( 69,232)( 70,231)( 71,233)( 72,229)( 73,228)( 74,230)( 75,245)( 76,244)( 77,243)( 78,242)( 79,241)( 80,240)( 81,239)( 82,238)( 83,237)( 84,120)( 85,122)( 86,121)( 87,126)( 88,128)( 89,127)( 90,123)( 91,125)( 92,124)( 93,130)( 94,129)( 95,131)( 96,136)( 97,135)( 98,137)( 99,133)(100,132)(101,134)(102,113)(103,112)(104,111)(105,119)(106,118)(107,117)(108,116)(109,115)(110,114)(138,144)(139,146)(140,145)(142,143)(147,154)(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);
s3 := 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, 59)( 31, 58)( 32, 57)( 33, 62)( 34, 61)( 35, 60)( 36, 65)( 37, 64)( 38, 63)( 39, 80)( 40, 79)( 41, 78)( 42, 83)( 43, 82)( 44, 81)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)( 50, 72)( 51, 68)( 52, 67)( 53, 66)( 54, 71)( 55, 70)( 56, 69)( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)( 99,102)(100,104)(101,103)(111,140)(112,139)(113,138)(114,143)(115,142)(116,141)(117,146)(118,145)(119,144)(120,161)(121,160)(122,159)(123,164)(124,163)(125,162)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)(132,149)(133,148)(134,147)(135,152)(136,151)(137,150)(166,167)(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)(181,185)(182,184)(192,221)(193,220)(194,219)(195,224)(196,223)(197,222)(198,227)(199,226)(200,225)(201,242)(202,241)(203,240)(204,245)(205,244)(206,243)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)(213,230)(214,229)(215,228)(216,233)(217,232)(218,231);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2 >;