Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,9,18}

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

Overview

Group
SmallGroup(1944,949)
Rank
4
Schläfli Type
{2,9,18}
Vertices, edges, …
2, 27, 243, 54
Order of s0s1s2s3
6
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,  8)(  9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)( 33, 62)( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)( 41, 76)( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)( 49, 68)( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)( 84,183)( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)( 92,191)( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)(100,180)(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)(108,172)(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)(116,240)(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)(124,232)(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)(132,224)(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)(140,211)(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)(148,203)(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)(156,192)(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)(164,200);;
s2 := (  3,195)(  4,197)(  5,196)(  6,199)(  7,198)(  8,200)(  9,194)( 10,193)( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)( 18,212)( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)( 26,209)( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)( 34,167)( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)( 42,183)( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)( 50,180)( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,220)( 58,219)( 59,221)( 60,224)( 61,223)( 62,222)( 63,225)( 64,227)( 65,226)( 66,238)( 67,237)( 68,239)( 69,242)( 70,241)( 71,240)( 72,243)( 73,245)( 74,244)( 75,229)( 76,228)( 77,230)( 78,233)( 79,232)( 80,231)( 81,234)( 82,236)( 83,235)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)( 90,113)( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)( 98,137)( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)(106,126)(107,128)(108,122)(109,121)(110,120)(138,139)(141,143)(145,146)(147,157)(148,156)(149,158)(150,161)(151,160)(152,159)(153,162)(154,164)(155,163);;
s3 := (  6, 10)(  7, 11)(  8,  9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)( 16, 29)( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)( 39, 48)( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)( 47, 52)( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)( 70, 83)( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 84,183)( 85,184)( 86,185)( 87,190)( 88,191)( 89,189)( 90,188)( 91,186)( 92,187)( 93,174)( 94,175)( 95,176)( 96,181)( 97,182)( 98,180)( 99,179)(100,177)(101,178)(102,165)(103,166)(104,167)(105,172)(106,173)(107,171)(108,170)(109,168)(110,169)(111,210)(112,211)(113,212)(114,217)(115,218)(116,216)(117,215)(118,213)(119,214)(120,201)(121,202)(122,203)(123,208)(124,209)(125,207)(126,206)(127,204)(128,205)(129,192)(130,193)(131,194)(132,199)(133,200)(134,198)(135,197)(136,195)(137,196)(138,237)(139,238)(140,239)(141,244)(142,245)(143,243)(144,242)(145,240)(146,241)(147,228)(148,229)(149,230)(150,235)(151,236)(152,234)(153,233)(154,231)(155,232)(156,219)(157,220)(158,221)(159,226)(160,227)(161,225)(162,224)(163,222)(164,223);;
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*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, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*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)!(  4,  5)(  6,  8)(  9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)( 33, 62)( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)( 41, 76)( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)( 49, 68)( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)( 84,183)( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)( 92,191)( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)(100,180)(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)(108,172)(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)(116,240)(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)(124,232)(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)(132,224)(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)(140,211)(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)(148,203)(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)(156,192)(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)(164,200);
s2 := Sym(245)!(  3,195)(  4,197)(  5,196)(  6,199)(  7,198)(  8,200)(  9,194)( 10,193)( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)( 18,212)( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)( 26,209)( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)( 34,167)( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)( 42,183)( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)( 50,180)( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,220)( 58,219)( 59,221)( 60,224)( 61,223)( 62,222)( 63,225)( 64,227)( 65,226)( 66,238)( 67,237)( 68,239)( 69,242)( 70,241)( 71,240)( 72,243)( 73,245)( 74,244)( 75,229)( 76,228)( 77,230)( 78,233)( 79,232)( 80,231)( 81,234)( 82,236)( 83,235)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)( 90,113)( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)( 98,137)( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)(106,126)(107,128)(108,122)(109,121)(110,120)(138,139)(141,143)(145,146)(147,157)(148,156)(149,158)(150,161)(151,160)(152,159)(153,162)(154,164)(155,163);
s3 := Sym(245)!(  6, 10)(  7, 11)(  8,  9)( 12, 21)( 13, 22)( 14, 23)( 15, 28)( 16, 29)( 17, 27)( 18, 26)( 19, 24)( 20, 25)( 33, 37)( 34, 38)( 35, 36)( 39, 48)( 40, 49)( 41, 50)( 42, 55)( 43, 56)( 44, 54)( 45, 53)( 46, 51)( 47, 52)( 60, 64)( 61, 65)( 62, 63)( 66, 75)( 67, 76)( 68, 77)( 69, 82)( 70, 83)( 71, 81)( 72, 80)( 73, 78)( 74, 79)( 84,183)( 85,184)( 86,185)( 87,190)( 88,191)( 89,189)( 90,188)( 91,186)( 92,187)( 93,174)( 94,175)( 95,176)( 96,181)( 97,182)( 98,180)( 99,179)(100,177)(101,178)(102,165)(103,166)(104,167)(105,172)(106,173)(107,171)(108,170)(109,168)(110,169)(111,210)(112,211)(113,212)(114,217)(115,218)(116,216)(117,215)(118,213)(119,214)(120,201)(121,202)(122,203)(123,208)(124,209)(125,207)(126,206)(127,204)(128,205)(129,192)(130,193)(131,194)(132,199)(133,200)(134,198)(135,197)(136,195)(137,196)(138,237)(139,238)(140,239)(141,244)(142,245)(143,243)(144,242)(145,240)(146,241)(147,228)(148,229)(149,230)(150,235)(151,236)(152,234)(153,233)(154,231)(155,232)(156,219)(157,220)(158,221)(159,226)(160,227)(161,225)(162,224)(163,222)(164,223);
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*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, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;