Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944e
if this polytope has a name.
Group : SmallGroup(1944,946)
Rank : 4
Schlafli Type : {2,18,9}
Number of vertices, edges, etc : 2, 54, 243, 27
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,6,9}*648d
   9-fold quotients : {2,6,3}*216
   27-fold quotients : {2,6,3}*72
   81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope