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Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944d
if this polytope has a name.
Group : SmallGroup(1944,945)
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, {2,18,3}*648
   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, 58)( 31, 57)( 32, 59)( 33, 61)
( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)( 41, 80)
( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)( 85, 86)
( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)
( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)(115,141)
(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)(123,163)
(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)(131,155)
(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)(169,170)
(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)
(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)(197,224)
(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)(205,243)
(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)(213,229)
(214,228)(215,230)(216,232)(217,231)(218,233);;
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, 58)( 60, 64)( 61, 63)( 62, 65)( 67, 68)
( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 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)(138,223)(139,222)(140,224)
(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)(148,233)
(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)(156,242)
(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)(164,243);;
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)( 30,160)( 31,159)( 32,161)( 33,157)( 34,156)
( 35,158)( 36,163)( 37,162)( 38,164)( 39,139)( 40,138)( 41,140)( 42,145)
( 43,144)( 44,146)( 45,142)( 46,141)( 47,143)( 48,154)( 49,153)( 50,155)
( 51,151)( 52,150)( 53,152)( 54,148)( 55,147)( 56,149)( 57,121)( 58,120)
( 59,122)( 60,127)( 61,126)( 62,128)( 63,124)( 64,123)( 65,125)( 66,136)
( 67,135)( 68,137)( 69,133)( 70,132)( 71,134)( 72,130)( 73,129)( 74,131)
( 75,115)( 76,114)( 77,116)( 78,112)( 79,111)( 80,113)( 81,118)( 82,117)
( 83,119)(165,168)(166,170)(167,169)(172,173)(175,176)(177,180)(178,182)
(179,181)(183,189)(184,191)(185,190)(187,188)(192,244)(193,243)(194,245)
(195,241)(196,240)(197,242)(198,238)(199,237)(200,239)(201,223)(202,222)
(203,224)(204,220)(205,219)(206,221)(207,226)(208,225)(209,227)(210,229)
(211,228)(212,230)(213,235)(214,234)(215,236)(216,232)(217,231)(218,233);;
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s3*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)(  7,  8)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 27)
( 16, 29)( 17, 28)( 18, 21)( 19, 23)( 20, 22)( 30, 58)( 31, 57)( 32, 59)
( 33, 61)( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)
( 41, 80)( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)
( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)
( 98,109)( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)
(115,141)(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)
(123,163)(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)
(131,155)(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)
(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)
(180,183)(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)
(197,224)(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)
(205,243)(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)
(213,229)(214,228)(215,230)(216,232)(217,231)(218,233);
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, 58)( 60, 64)( 61, 63)( 62, 65)
( 67, 68)( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)
( 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)(138,223)(139,222)
(140,224)(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)
(148,233)(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)
(156,242)(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)
(164,243);
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)( 30,160)( 31,159)( 32,161)( 33,157)
( 34,156)( 35,158)( 36,163)( 37,162)( 38,164)( 39,139)( 40,138)( 41,140)
( 42,145)( 43,144)( 44,146)( 45,142)( 46,141)( 47,143)( 48,154)( 49,153)
( 50,155)( 51,151)( 52,150)( 53,152)( 54,148)( 55,147)( 56,149)( 57,121)
( 58,120)( 59,122)( 60,127)( 61,126)( 62,128)( 63,124)( 64,123)( 65,125)
( 66,136)( 67,135)( 68,137)( 69,133)( 70,132)( 71,134)( 72,130)( 73,129)
( 74,131)( 75,115)( 76,114)( 77,116)( 78,112)( 79,111)( 80,113)( 81,118)
( 82,117)( 83,119)(165,168)(166,170)(167,169)(172,173)(175,176)(177,180)
(178,182)(179,181)(183,189)(184,191)(185,190)(187,188)(192,244)(193,243)
(194,245)(195,241)(196,240)(197,242)(198,238)(199,237)(200,239)(201,223)
(202,222)(203,224)(204,220)(205,219)(206,221)(207,226)(208,225)(209,227)
(210,229)(211,228)(212,230)(213,235)(214,234)(215,236)(216,232)(217,231)
(218,233);
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*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*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*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

to this polytope