Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,940)
Rank : 4
Schlafli Type : {2,9,18}
Number of vertices, edges, etc : 2, 27, 243, 54
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 : {2,9,18}*648, {2,9,6}*648a
9-fold quotients : {2,9,6}*216, {2,3,6}*216
27-fold quotients : {2,9,2}*72, {2,3,6}*72
81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 6, 9)( 7, 10)( 8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 30, 63)( 31, 64)( 32, 65)( 33, 60)
( 34, 61)( 35, 62)( 36, 57)( 37, 58)( 38, 59)( 39, 81)( 40, 82)( 41, 83)
( 42, 78)( 43, 79)( 44, 80)( 45, 75)( 46, 76)( 47, 77)( 48, 72)( 49, 73)
( 50, 74)( 51, 69)( 52, 70)( 53, 71)( 54, 66)( 55, 67)( 56, 68)( 84,183)
( 85,184)( 86,185)( 87,189)( 88,190)( 89,191)( 90,186)( 91,187)( 92,188)
( 93,174)( 94,175)( 95,176)( 96,180)( 97,181)( 98,182)( 99,177)(100,178)
(101,179)(102,165)(103,166)(104,167)(105,171)(106,172)(107,173)(108,168)
(109,169)(110,170)(111,243)(112,244)(113,245)(114,240)(115,241)(116,242)
(117,237)(118,238)(119,239)(120,234)(121,235)(122,236)(123,231)(124,232)
(125,233)(126,228)(127,229)(128,230)(129,225)(130,226)(131,227)(132,222)
(133,223)(134,224)(135,219)(136,220)(137,221)(138,216)(139,217)(140,218)
(141,213)(142,214)(143,215)(144,210)(145,211)(146,212)(147,207)(148,208)
(149,209)(150,204)(151,205)(152,206)(153,201)(154,202)(155,203)(156,198)
(157,199)(158,200)(159,195)(160,196)(161,197)(162,192)(163,193)(164,194);;
s2 := ( 3,193)( 4,194)( 5,192)( 6,199)( 7,200)( 8,198)( 9,196)( 10,197)
( 11,195)( 12,211)( 13,212)( 14,210)( 15,217)( 16,218)( 17,216)( 18,214)
( 19,215)( 20,213)( 21,202)( 22,203)( 23,201)( 24,208)( 25,209)( 26,207)
( 27,205)( 28,206)( 29,204)( 30,167)( 31,165)( 32,166)( 33,173)( 34,171)
( 35,172)( 36,170)( 37,168)( 38,169)( 39,185)( 40,183)( 41,184)( 42,191)
( 43,189)( 44,190)( 45,188)( 46,186)( 47,187)( 48,176)( 49,174)( 50,175)
( 51,182)( 52,180)( 53,181)( 54,179)( 55,177)( 56,178)( 57,225)( 58,226)
( 59,227)( 60,222)( 61,223)( 62,224)( 63,219)( 64,220)( 65,221)( 66,243)
( 67,244)( 68,245)( 69,240)( 70,241)( 71,242)( 72,237)( 73,238)( 74,239)
( 75,234)( 76,235)( 77,236)( 78,231)( 79,232)( 80,233)( 81,228)( 82,229)
( 83,230)( 84,112)( 85,113)( 86,111)( 87,118)( 88,119)( 89,117)( 90,115)
( 91,116)( 92,114)( 93,130)( 94,131)( 95,129)( 96,136)( 97,137)( 98,135)
( 99,133)(100,134)(101,132)(102,121)(103,122)(104,120)(105,127)(106,128)
(107,126)(108,124)(109,125)(110,123)(138,144)(139,145)(140,146)(147,162)
(148,163)(149,164)(150,159)(151,160)(152,161)(153,156)(154,157)(155,158);;
s3 := ( 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);;
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*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*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)!( 6, 9)( 7, 10)( 8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 30, 63)( 31, 64)( 32, 65)
( 33, 60)( 34, 61)( 35, 62)( 36, 57)( 37, 58)( 38, 59)( 39, 81)( 40, 82)
( 41, 83)( 42, 78)( 43, 79)( 44, 80)( 45, 75)( 46, 76)( 47, 77)( 48, 72)
( 49, 73)( 50, 74)( 51, 69)( 52, 70)( 53, 71)( 54, 66)( 55, 67)( 56, 68)
( 84,183)( 85,184)( 86,185)( 87,189)( 88,190)( 89,191)( 90,186)( 91,187)
( 92,188)( 93,174)( 94,175)( 95,176)( 96,180)( 97,181)( 98,182)( 99,177)
(100,178)(101,179)(102,165)(103,166)(104,167)(105,171)(106,172)(107,173)
(108,168)(109,169)(110,170)(111,243)(112,244)(113,245)(114,240)(115,241)
(116,242)(117,237)(118,238)(119,239)(120,234)(121,235)(122,236)(123,231)
(124,232)(125,233)(126,228)(127,229)(128,230)(129,225)(130,226)(131,227)
(132,222)(133,223)(134,224)(135,219)(136,220)(137,221)(138,216)(139,217)
(140,218)(141,213)(142,214)(143,215)(144,210)(145,211)(146,212)(147,207)
(148,208)(149,209)(150,204)(151,205)(152,206)(153,201)(154,202)(155,203)
(156,198)(157,199)(158,200)(159,195)(160,196)(161,197)(162,192)(163,193)
(164,194);
s2 := Sym(245)!( 3,193)( 4,194)( 5,192)( 6,199)( 7,200)( 8,198)( 9,196)
( 10,197)( 11,195)( 12,211)( 13,212)( 14,210)( 15,217)( 16,218)( 17,216)
( 18,214)( 19,215)( 20,213)( 21,202)( 22,203)( 23,201)( 24,208)( 25,209)
( 26,207)( 27,205)( 28,206)( 29,204)( 30,167)( 31,165)( 32,166)( 33,173)
( 34,171)( 35,172)( 36,170)( 37,168)( 38,169)( 39,185)( 40,183)( 41,184)
( 42,191)( 43,189)( 44,190)( 45,188)( 46,186)( 47,187)( 48,176)( 49,174)
( 50,175)( 51,182)( 52,180)( 53,181)( 54,179)( 55,177)( 56,178)( 57,225)
( 58,226)( 59,227)( 60,222)( 61,223)( 62,224)( 63,219)( 64,220)( 65,221)
( 66,243)( 67,244)( 68,245)( 69,240)( 70,241)( 71,242)( 72,237)( 73,238)
( 74,239)( 75,234)( 76,235)( 77,236)( 78,231)( 79,232)( 80,233)( 81,228)
( 82,229)( 83,230)( 84,112)( 85,113)( 86,111)( 87,118)( 88,119)( 89,117)
( 90,115)( 91,116)( 92,114)( 93,130)( 94,131)( 95,129)( 96,136)( 97,137)
( 98,135)( 99,133)(100,134)(101,132)(102,121)(103,122)(104,120)(105,127)
(106,128)(107,126)(108,124)(109,125)(110,123)(138,144)(139,145)(140,146)
(147,162)(148,163)(149,164)(150,159)(151,160)(152,161)(153,156)(154,157)
(155,158);
s3 := 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);
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*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope