Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944i
if this polytope has a name.
Group : SmallGroup(1944,951)
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 : {2,9,6}*648a, {2,3,18}*648
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 := ( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)
( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 58)( 31, 57)( 32, 59)( 33, 60)
( 34, 62)( 35, 61)( 36, 65)( 37, 64)( 38, 63)( 39, 76)( 40, 75)( 41, 77)
( 42, 78)( 43, 80)( 44, 79)( 45, 83)( 46, 82)( 47, 81)( 48, 67)( 49, 66)
( 50, 68)( 51, 69)( 52, 71)( 53, 70)( 54, 74)( 55, 73)( 56, 72)( 84,183)
( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)( 92,191)
( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)(100,180)
(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)(108,172)
(109,171)(110,173)(111,238)(112,237)(113,239)(114,240)(115,242)(116,241)
(117,245)(118,244)(119,243)(120,229)(121,228)(122,230)(123,231)(124,233)
(125,232)(126,236)(127,235)(128,234)(129,220)(130,219)(131,221)(132,222)
(133,224)(134,223)(135,227)(136,226)(137,225)(138,211)(139,210)(140,212)
(141,213)(142,215)(143,214)(144,218)(145,217)(146,216)(147,202)(148,201)
(149,203)(150,204)(151,206)(152,205)(153,209)(154,208)(155,207)(156,193)
(157,192)(158,194)(159,195)(160,197)(161,196)(162,200)(163,199)(164,198);;
s2 := ( 3,195)( 4,197)( 5,196)( 6,199)( 7,198)( 8,200)( 9,194)( 10,193)
( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)( 18,212)
( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)( 26,209)
( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)( 34,167)
( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)( 42,183)
( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)( 50,180)
( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,221)( 58,220)
( 59,219)( 60,222)( 61,224)( 62,223)( 63,226)( 64,225)( 65,227)( 66,239)
( 67,238)( 68,237)( 69,240)( 70,242)( 71,241)( 72,244)( 73,243)( 74,245)
( 75,230)( 76,229)( 77,228)( 78,231)( 79,233)( 80,232)( 81,235)( 82,234)
( 83,236)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)( 90,113)
( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)( 98,137)
( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)(106,126)
(107,128)(108,122)(109,121)(110,120)(138,140)(142,143)(144,145)(147,158)
(148,157)(149,156)(150,159)(151,161)(152,160)(153,163)(154,162)(155,164);;
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, 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);;
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*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)
( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 58)( 31, 57)( 32, 59)
( 33, 60)( 34, 62)( 35, 61)( 36, 65)( 37, 64)( 38, 63)( 39, 76)( 40, 75)
( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 83)( 46, 82)( 47, 81)( 48, 67)
( 49, 66)( 50, 68)( 51, 69)( 52, 71)( 53, 70)( 54, 74)( 55, 73)( 56, 72)
( 84,183)( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)
( 92,191)( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)
(100,180)(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)
(108,172)(109,171)(110,173)(111,238)(112,237)(113,239)(114,240)(115,242)
(116,241)(117,245)(118,244)(119,243)(120,229)(121,228)(122,230)(123,231)
(124,233)(125,232)(126,236)(127,235)(128,234)(129,220)(130,219)(131,221)
(132,222)(133,224)(134,223)(135,227)(136,226)(137,225)(138,211)(139,210)
(140,212)(141,213)(142,215)(143,214)(144,218)(145,217)(146,216)(147,202)
(148,201)(149,203)(150,204)(151,206)(152,205)(153,209)(154,208)(155,207)
(156,193)(157,192)(158,194)(159,195)(160,197)(161,196)(162,200)(163,199)
(164,198);
s2 := Sym(245)!( 3,195)( 4,197)( 5,196)( 6,199)( 7,198)( 8,200)( 9,194)
( 10,193)( 11,192)( 12,213)( 13,215)( 14,214)( 15,217)( 16,216)( 17,218)
( 18,212)( 19,211)( 20,210)( 21,204)( 22,206)( 23,205)( 24,208)( 25,207)
( 26,209)( 27,203)( 28,202)( 29,201)( 30,173)( 31,172)( 32,171)( 33,165)
( 34,167)( 35,166)( 36,169)( 37,168)( 38,170)( 39,191)( 40,190)( 41,189)
( 42,183)( 43,185)( 44,184)( 45,187)( 46,186)( 47,188)( 48,182)( 49,181)
( 50,180)( 51,174)( 52,176)( 53,175)( 54,178)( 55,177)( 56,179)( 57,221)
( 58,220)( 59,219)( 60,222)( 61,224)( 62,223)( 63,226)( 64,225)( 65,227)
( 66,239)( 67,238)( 68,237)( 69,240)( 70,242)( 71,241)( 72,244)( 73,243)
( 74,245)( 75,230)( 76,229)( 77,228)( 78,231)( 79,233)( 80,232)( 81,235)
( 82,234)( 83,236)( 84,114)( 85,116)( 86,115)( 87,118)( 88,117)( 89,119)
( 90,113)( 91,112)( 92,111)( 93,132)( 94,134)( 95,133)( 96,136)( 97,135)
( 98,137)( 99,131)(100,130)(101,129)(102,123)(103,125)(104,124)(105,127)
(106,126)(107,128)(108,122)(109,121)(110,120)(138,140)(142,143)(144,145)
(147,158)(148,157)(149,156)(150,159)(151,161)(152,160)(153,163)(154,162)
(155,164);
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, 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);
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*s1*s2*s1*s2*s3*s2*s1*s2,
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 >;
to this polytope