Polytope of Type {2,18,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,9}*1944h
if this polytope has a name.
Group : SmallGroup(1944,950)
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
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, 59)( 31, 58)( 32, 57)( 33, 65)
( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)( 41, 66)
( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)( 49, 76)
( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)(115,145)
(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)(123,155)
(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)(131,156)
(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,221)(193,220)(194,219)(195,227)(196,226)(197,225)
(198,224)(199,223)(200,222)(201,230)(202,229)(203,228)(204,236)(205,235)
(206,234)(207,233)(208,232)(209,231)(210,239)(211,238)(212,237)(213,245)
(214,244)(215,243)(216,242)(217,241)(218,240);;
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, 59)( 60, 61)( 64, 65)( 66, 77)( 67, 76)
( 68, 75)( 69, 79)( 70, 78)( 71, 80)( 72, 81)( 73, 83)( 74, 82)( 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)(138,239)(139,238)(140,237)
(141,241)(142,240)(143,242)(144,243)(145,245)(146,244)(147,230)(148,229)
(149,228)(150,232)(151,231)(152,233)(153,234)(154,236)(155,235)(156,221)
(157,220)(158,219)(159,223)(160,222)(161,224)(162,225)(163,227)(164,226);;
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)( 30,146)( 31,145)( 32,144)( 33,140)( 34,139)
( 35,138)( 36,143)( 37,142)( 38,141)( 39,164)( 40,163)( 41,162)( 42,158)
( 43,157)( 44,156)( 45,161)( 46,160)( 47,159)( 48,155)( 49,154)( 50,153)
( 51,149)( 52,148)( 53,147)( 54,152)( 55,151)( 56,150)( 57,116)( 58,115)
( 59,114)( 60,119)( 61,118)( 62,117)( 63,113)( 64,112)( 65,111)( 66,134)
( 67,133)( 68,132)( 69,137)( 70,136)( 71,135)( 72,131)( 73,130)( 74,129)
( 75,125)( 76,124)( 77,123)( 78,128)( 79,127)( 80,126)( 81,122)( 82,121)
( 83,120)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)(171,189)
(172,191)(173,190)(175,176)(178,179)(181,182)(192,245)(193,244)(194,243)
(195,239)(196,238)(197,237)(198,242)(199,241)(200,240)(201,236)(202,235)
(203,234)(204,230)(205,229)(206,228)(207,233)(208,232)(209,231)(210,227)
(211,226)(212,225)(213,221)(214,220)(215,219)(216,224)(217,223)(218,222);;
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*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
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,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*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, 59)( 31, 58)( 32, 57)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 68)( 40, 67)
( 41, 66)( 42, 74)( 43, 73)( 44, 72)( 45, 71)( 46, 70)( 47, 69)( 48, 77)
( 49, 76)( 50, 75)( 51, 83)( 52, 82)( 53, 81)( 54, 80)( 55, 79)( 56, 78)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,140)(112,139)(113,138)(114,146)
(115,145)(116,144)(117,143)(118,142)(119,141)(120,149)(121,148)(122,147)
(123,155)(124,154)(125,153)(126,152)(127,151)(128,150)(129,158)(130,157)
(131,156)(132,164)(133,163)(134,162)(135,161)(136,160)(137,159)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,221)(193,220)(194,219)(195,227)(196,226)
(197,225)(198,224)(199,223)(200,222)(201,230)(202,229)(203,228)(204,236)
(205,235)(206,234)(207,233)(208,232)(209,231)(210,239)(211,238)(212,237)
(213,245)(214,244)(215,243)(216,242)(217,241)(218,240);
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, 59)( 60, 61)( 64, 65)( 66, 77)
( 67, 76)( 68, 75)( 69, 79)( 70, 78)( 71, 80)( 72, 81)( 73, 83)( 74, 82)
( 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)(138,239)(139,238)
(140,237)(141,241)(142,240)(143,242)(144,243)(145,245)(146,244)(147,230)
(148,229)(149,228)(150,232)(151,231)(152,233)(153,234)(154,236)(155,235)
(156,221)(157,220)(158,219)(159,223)(160,222)(161,224)(162,225)(163,227)
(164,226);
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)( 30,146)( 31,145)( 32,144)( 33,140)
( 34,139)( 35,138)( 36,143)( 37,142)( 38,141)( 39,164)( 40,163)( 41,162)
( 42,158)( 43,157)( 44,156)( 45,161)( 46,160)( 47,159)( 48,155)( 49,154)
( 50,153)( 51,149)( 52,148)( 53,147)( 54,152)( 55,151)( 56,150)( 57,116)
( 58,115)( 59,114)( 60,119)( 61,118)( 62,117)( 63,113)( 64,112)( 65,111)
( 66,134)( 67,133)( 68,132)( 69,137)( 70,136)( 71,135)( 72,131)( 73,130)
( 74,129)( 75,125)( 76,124)( 77,123)( 78,128)( 79,127)( 80,126)( 81,122)
( 82,121)( 83,120)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)
(171,189)(172,191)(173,190)(175,176)(178,179)(181,182)(192,245)(193,244)
(194,243)(195,239)(196,238)(197,237)(198,242)(199,241)(200,240)(201,236)
(202,235)(203,234)(204,230)(205,229)(206,228)(207,233)(208,232)(209,231)
(210,227)(211,226)(212,225)(213,221)(214,220)(215,219)(216,224)(217,223)
(218,222);
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*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
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,
s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2 >;
to this polytope