Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,18,9}

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

Overview

Group
SmallGroup(1944,952)
Rank
4
Schläfli Type
{2,18,9}
Vertices, edges, …
2, 54, 243, 27
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,  9)(  7, 11)(  8, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 27)( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)( 34, 38)( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)( 45, 51)( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)( 67, 77)( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)( 84,183)( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)( 92,187)( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)(100,179)(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)(108,168)(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)(116,217)(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)(124,209)(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)(132,198)(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)(140,238)(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)(148,230)(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)(156,219)(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)(164,223);;
s2 := (  3, 84)(  4, 86)(  5, 85)(  6, 89)(  7, 88)(  8, 87)(  9, 91)( 10, 90)( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)( 18,109)( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)( 26, 96)( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)( 34,140)( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)( 42,156)( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)( 50,155)( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)( 58,116)( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)( 66,132)( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)( 74,131)( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)(171,190)(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)(194,245)(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)(202,234)(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)(210,226)(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)(218,222);;
s3 := (  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)( 10, 38)( 11, 37)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 53)( 17, 52)( 18, 54)( 19, 56)( 20, 55)( 21, 39)( 22, 41)( 23, 40)( 24, 42)( 25, 44)( 26, 43)( 27, 45)( 28, 47)( 29, 46)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 84,210)( 85,212)( 86,211)( 87,213)( 88,215)( 89,214)( 90,216)( 91,218)( 92,217)( 93,201)( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)(100,209)(101,208)(102,192)(103,194)(104,193)(105,195)(106,197)(107,196)(108,198)(109,200)(110,199)(111,183)(112,185)(113,184)(114,186)(115,188)(116,187)(117,189)(118,191)(119,190)(120,174)(121,176)(122,175)(123,177)(124,179)(125,178)(126,180)(127,182)(128,181)(129,165)(130,167)(131,166)(132,168)(133,170)(134,169)(135,171)(136,173)(137,172)(138,237)(139,239)(140,238)(141,240)(142,242)(143,241)(144,243)(145,245)(146,244)(147,228)(148,230)(149,229)(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)(156,219)(157,221)(158,220)(159,222)(160,224)(161,223)(162,225)(163,227)(164,226);;
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*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 27)( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)( 34, 38)( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)( 45, 51)( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)( 67, 77)( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)( 84,183)( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)( 92,187)( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)(100,179)(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)(108,168)(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)(116,217)(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)(124,209)(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)(132,198)(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)(140,238)(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)(148,230)(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)(156,219)(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)(164,223);
s2 := Sym(245)!(  3, 84)(  4, 86)(  5, 85)(  6, 89)(  7, 88)(  8, 87)(  9, 91)( 10, 90)( 11, 92)( 12,102)( 13,104)( 14,103)( 15,107)( 16,106)( 17,105)( 18,109)( 19,108)( 20,110)( 21, 93)( 22, 95)( 23, 94)( 24, 98)( 25, 97)( 26, 96)( 27,100)( 28, 99)( 29,101)( 30,145)( 31,144)( 32,146)( 33,138)( 34,140)( 35,139)( 36,143)( 37,142)( 38,141)( 39,163)( 40,162)( 41,164)( 42,156)( 43,158)( 44,157)( 45,161)( 46,160)( 47,159)( 48,154)( 49,153)( 50,155)( 51,147)( 52,149)( 53,148)( 54,152)( 55,151)( 56,150)( 57,114)( 58,116)( 59,115)( 60,119)( 61,118)( 62,117)( 63,112)( 64,111)( 65,113)( 66,132)( 67,134)( 68,133)( 69,137)( 70,136)( 71,135)( 72,130)( 73,129)( 74,131)( 75,123)( 76,125)( 77,124)( 78,128)( 79,127)( 80,126)( 81,121)( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,188)(169,187)(170,186)(171,190)(172,189)(173,191)(175,176)(177,179)(180,181)(192,244)(193,243)(194,245)(195,237)(196,239)(197,238)(198,242)(199,241)(200,240)(201,235)(202,234)(203,236)(204,228)(205,230)(206,229)(207,233)(208,232)(209,231)(210,226)(211,225)(212,227)(213,219)(214,221)(215,220)(216,224)(217,223)(218,222);
s3 := Sym(245)!(  3, 30)(  4, 32)(  5, 31)(  6, 33)(  7, 35)(  8, 34)(  9, 36)( 10, 38)( 11, 37)( 12, 48)( 13, 50)( 14, 49)( 15, 51)( 16, 53)( 17, 52)( 18, 54)( 19, 56)( 20, 55)( 21, 39)( 22, 41)( 23, 40)( 24, 42)( 25, 44)( 26, 43)( 27, 45)( 28, 47)( 29, 46)( 58, 59)( 61, 62)( 64, 65)( 66, 75)( 67, 77)( 68, 76)( 69, 78)( 70, 80)( 71, 79)( 72, 81)( 73, 83)( 74, 82)( 84,210)( 85,212)( 86,211)( 87,213)( 88,215)( 89,214)( 90,216)( 91,218)( 92,217)( 93,201)( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)(100,209)(101,208)(102,192)(103,194)(104,193)(105,195)(106,197)(107,196)(108,198)(109,200)(110,199)(111,183)(112,185)(113,184)(114,186)(115,188)(116,187)(117,189)(118,191)(119,190)(120,174)(121,176)(122,175)(123,177)(124,179)(125,178)(126,180)(127,182)(128,181)(129,165)(130,167)(131,166)(132,168)(133,170)(134,169)(135,171)(136,173)(137,172)(138,237)(139,239)(140,238)(141,240)(142,242)(143,241)(144,243)(145,245)(146,244)(147,228)(148,230)(149,229)(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)(156,219)(157,221)(158,220)(159,222)(160,224)(161,223)(162,225)(163,227)(164,226);
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*s1*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2, 
s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2 >;