Polytope of Type {2,18,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944i
if this polytope has a name.
Group : SmallGroup(1944,951)
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}*648a, {2,18,3}*648
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, 58)( 31, 57)( 32, 59)( 33, 64)
( 34, 63)( 35, 65)( 36, 61)( 37, 60)( 38, 62)( 39, 67)( 40, 66)( 41, 68)
( 42, 73)( 43, 72)( 44, 74)( 45, 70)( 46, 69)( 47, 71)( 48, 76)( 49, 75)
( 50, 77)( 51, 82)( 52, 81)( 53, 83)( 54, 79)( 55, 78)( 56, 80)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,139)(112,138)(113,140)(114,145)(115,144)
(116,146)(117,142)(118,141)(119,143)(120,148)(121,147)(122,149)(123,154)
(124,153)(125,155)(126,151)(127,150)(128,152)(129,157)(130,156)(131,158)
(132,163)(133,162)(134,164)(135,160)(136,159)(137,161)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,220)(193,219)(194,221)(195,226)(196,225)(197,227)
(198,223)(199,222)(200,224)(201,229)(202,228)(203,230)(204,235)(205,234)
(206,236)(207,232)(208,231)(209,233)(210,238)(211,237)(212,239)(213,244)
(214,243)(215,245)(216,241)(217,240)(218,242);;
s2 := ( 3, 30)( 4, 32)( 5, 31)( 6, 35)( 7, 34)( 8, 33)( 9, 37)( 10, 36)
( 11, 38)( 12, 48)( 13, 50)( 14, 49)( 15, 53)( 16, 52)( 17, 51)( 18, 55)
( 19, 54)( 20, 56)( 21, 39)( 22, 41)( 23, 40)( 24, 44)( 25, 43)( 26, 42)
( 27, 46)( 28, 45)( 29, 47)( 57, 58)( 61, 62)( 63, 65)( 66, 76)( 67, 75)
( 68, 77)( 69, 78)( 70, 80)( 71, 79)( 72, 83)( 73, 82)( 74, 81)( 84,210)
( 85,212)( 86,211)( 87,215)( 88,214)( 89,213)( 90,217)( 91,216)( 92,218)
( 93,201)( 94,203)( 95,202)( 96,206)( 97,205)( 98,204)( 99,208)(100,207)
(101,209)(102,192)(103,194)(104,193)(105,197)(106,196)(107,195)(108,199)
(109,198)(110,200)(111,183)(112,185)(113,184)(114,188)(115,187)(116,186)
(117,190)(118,189)(119,191)(120,174)(121,176)(122,175)(123,179)(124,178)
(125,177)(126,181)(127,180)(128,182)(129,165)(130,167)(131,166)(132,170)
(133,169)(134,168)(135,172)(136,171)(137,173)(138,238)(139,237)(140,239)
(141,240)(142,242)(143,241)(144,245)(145,244)(146,243)(147,229)(148,228)
(149,230)(150,231)(151,233)(152,232)(153,236)(154,235)(155,234)(156,220)
(157,219)(158,221)(159,222)(160,224)(161,223)(162,227)(163,226)(164,225);;
s3 := ( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)( 10, 92)
( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)
( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)
( 27, 99)( 28,101)( 29,100)( 30,145)( 31,144)( 32,146)( 33,139)( 34,138)
( 35,140)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)( 42,157)
( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,154)( 49,153)( 50,155)
( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,115)( 58,114)
( 59,116)( 60,118)( 61,117)( 62,119)( 63,112)( 64,111)( 65,113)( 66,133)
( 67,132)( 68,134)( 69,136)( 70,135)( 71,137)( 72,130)( 73,129)( 74,131)
( 75,124)( 76,123)( 77,125)( 78,127)( 79,126)( 80,128)( 81,121)( 82,120)
( 83,122)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)(171,189)
(172,191)(173,190)(175,176)(178,179)(181,182)(192,244)(193,243)(194,245)
(195,238)(196,237)(197,239)(198,241)(199,240)(200,242)(201,235)(202,234)
(203,236)(204,229)(205,228)(206,230)(207,232)(208,231)(209,233)(210,226)
(211,225)(212,227)(213,220)(214,219)(215,221)(216,223)(217,222)(218,224);;
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 58)( 31, 57)( 32, 59)
( 33, 64)( 34, 63)( 35, 65)( 36, 61)( 37, 60)( 38, 62)( 39, 67)( 40, 66)
( 41, 68)( 42, 73)( 43, 72)( 44, 74)( 45, 70)( 46, 69)( 47, 71)( 48, 76)
( 49, 75)( 50, 77)( 51, 82)( 52, 81)( 53, 83)( 54, 79)( 55, 78)( 56, 80)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,139)(112,138)(113,140)(114,145)
(115,144)(116,146)(117,142)(118,141)(119,143)(120,148)(121,147)(122,149)
(123,154)(124,153)(125,155)(126,151)(127,150)(128,152)(129,157)(130,156)
(131,158)(132,163)(133,162)(134,164)(135,160)(136,159)(137,161)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,220)(193,219)(194,221)(195,226)(196,225)
(197,227)(198,223)(199,222)(200,224)(201,229)(202,228)(203,230)(204,235)
(205,234)(206,236)(207,232)(208,231)(209,233)(210,238)(211,237)(212,239)
(213,244)(214,243)(215,245)(216,241)(217,240)(218,242);
s2 := Sym(245)!( 3, 30)( 4, 32)( 5, 31)( 6, 35)( 7, 34)( 8, 33)( 9, 37)
( 10, 36)( 11, 38)( 12, 48)( 13, 50)( 14, 49)( 15, 53)( 16, 52)( 17, 51)
( 18, 55)( 19, 54)( 20, 56)( 21, 39)( 22, 41)( 23, 40)( 24, 44)( 25, 43)
( 26, 42)( 27, 46)( 28, 45)( 29, 47)( 57, 58)( 61, 62)( 63, 65)( 66, 76)
( 67, 75)( 68, 77)( 69, 78)( 70, 80)( 71, 79)( 72, 83)( 73, 82)( 74, 81)
( 84,210)( 85,212)( 86,211)( 87,215)( 88,214)( 89,213)( 90,217)( 91,216)
( 92,218)( 93,201)( 94,203)( 95,202)( 96,206)( 97,205)( 98,204)( 99,208)
(100,207)(101,209)(102,192)(103,194)(104,193)(105,197)(106,196)(107,195)
(108,199)(109,198)(110,200)(111,183)(112,185)(113,184)(114,188)(115,187)
(116,186)(117,190)(118,189)(119,191)(120,174)(121,176)(122,175)(123,179)
(124,178)(125,177)(126,181)(127,180)(128,182)(129,165)(130,167)(131,166)
(132,170)(133,169)(134,168)(135,172)(136,171)(137,173)(138,238)(139,237)
(140,239)(141,240)(142,242)(143,241)(144,245)(145,244)(146,243)(147,229)
(148,228)(149,230)(150,231)(151,233)(152,232)(153,236)(154,235)(155,234)
(156,220)(157,219)(158,221)(159,222)(160,224)(161,223)(162,227)(163,226)
(164,225);
s3 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)
( 10, 92)( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)
( 18,108)( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)
( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,145)( 31,144)( 32,146)( 33,139)
( 34,138)( 35,140)( 36,142)( 37,141)( 38,143)( 39,163)( 40,162)( 41,164)
( 42,157)( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,154)( 49,153)
( 50,155)( 51,148)( 52,147)( 53,149)( 54,151)( 55,150)( 56,152)( 57,115)
( 58,114)( 59,116)( 60,118)( 61,117)( 62,119)( 63,112)( 64,111)( 65,113)
( 66,133)( 67,132)( 68,134)( 69,136)( 70,135)( 71,137)( 72,130)( 73,129)
( 74,131)( 75,124)( 76,123)( 77,125)( 78,127)( 79,126)( 80,128)( 81,121)
( 82,120)( 83,122)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)
(171,189)(172,191)(173,190)(175,176)(178,179)(181,182)(192,244)(193,243)
(194,245)(195,238)(196,237)(197,239)(198,241)(199,240)(200,242)(201,235)
(202,234)(203,236)(204,229)(205,228)(206,230)(207,232)(208,231)(209,233)
(210,226)(211,225)(212,227)(213,220)(214,219)(215,221)(216,223)(217,222)
(218,224);
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,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*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