Part of the Atlas of Small Regular Polytopes

Polytope of Type {2,6,18}

Atlas Canonical Name {2,6,18}*1944d

Overview

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