Polytope of Type {2,18,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944a
if this polytope has a name.
Group : SmallGroup(1944,940)
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 : {2,18,9}*648, {2,6,9}*648a
   9-fold quotients : {2,6,9}*216, {2,6,3}*216
   27-fold quotients : {2,2,9}*72, {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)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)(197,223)
(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)(205,233)
(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)(213,240)
(214,242)(215,241)(216,237)(217,239)(218,238);;
s2 := (  3, 30)(  4, 31)(  5, 32)(  6, 36)(  7, 37)(  8, 38)(  9, 33)( 10, 34)
( 11, 35)( 12, 48)( 13, 49)( 14, 50)( 15, 54)( 16, 55)( 17, 56)( 18, 51)
( 19, 52)( 20, 53)( 21, 39)( 22, 40)( 23, 41)( 24, 45)( 25, 46)( 26, 47)
( 27, 42)( 28, 43)( 29, 44)( 57, 63)( 58, 64)( 59, 65)( 66, 81)( 67, 82)
( 68, 83)( 69, 78)( 70, 79)( 71, 80)( 72, 75)( 73, 76)( 74, 77)( 84,210)
( 85,211)( 86,212)( 87,216)( 88,217)( 89,218)( 90,213)( 91,214)( 92,215)
( 93,201)( 94,202)( 95,203)( 96,207)( 97,208)( 98,209)( 99,204)(100,205)
(101,206)(102,192)(103,193)(104,194)(105,198)(106,199)(107,200)(108,195)
(109,196)(110,197)(111,183)(112,184)(113,185)(114,189)(115,190)(116,191)
(117,186)(118,187)(119,188)(120,174)(121,175)(122,176)(123,180)(124,181)
(125,182)(126,177)(127,178)(128,179)(129,165)(130,166)(131,167)(132,171)
(133,172)(134,173)(135,168)(136,169)(137,170)(138,243)(139,244)(140,245)
(141,240)(142,241)(143,242)(144,237)(145,238)(146,239)(147,234)(148,235)
(149,236)(150,231)(151,232)(152,233)(153,228)(154,229)(155,230)(156,225)
(157,226)(158,227)(159,222)(160,223)(161,224)(162,219)(163,220)(164,221);;
s3 := (  3, 84)(  4, 85)(  5, 86)(  6, 90)(  7, 91)(  8, 92)(  9, 87)( 10, 88)
( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)( 18,105)
( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)( 26,101)
( 27, 96)( 28, 97)( 29, 98)( 30,146)( 31,144)( 32,145)( 33,143)( 34,141)
( 35,142)( 36,140)( 37,138)( 38,139)( 39,164)( 40,162)( 41,163)( 42,161)
( 43,159)( 44,160)( 45,158)( 46,156)( 47,157)( 48,155)( 49,153)( 50,154)
( 51,152)( 52,150)( 53,151)( 54,149)( 55,147)( 56,148)( 57,118)( 58,119)
( 59,117)( 60,115)( 61,116)( 62,114)( 63,112)( 64,113)( 65,111)( 66,136)
( 67,137)( 68,135)( 69,133)( 70,134)( 71,132)( 72,130)( 73,131)( 74,129)
( 75,127)( 76,128)( 77,126)( 78,124)( 79,125)( 80,123)( 81,121)( 82,122)
( 83,120)(165,183)(166,184)(167,185)(168,189)(169,190)(170,191)(171,186)
(172,187)(173,188)(177,180)(178,181)(179,182)(192,245)(193,243)(194,244)
(195,242)(196,240)(197,241)(198,239)(199,237)(200,238)(201,236)(202,234)
(203,235)(204,233)(205,231)(206,232)(207,230)(208,228)(209,229)(210,227)
(211,225)(212,226)(213,224)(214,222)(215,223)(216,221)(217,219)(218,220);;
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*s1*s2*s3*s2*s3*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope