Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944h
if this polytope has a name.
Group : SmallGroup(1944,950)
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
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, 59)( 31, 58)( 32, 57)( 33, 61)
( 34, 60)( 35, 62)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)( 41, 75)
( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)( 49, 67)
( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)( 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)(111,239)(112,238)(113,237)(114,241)(115,240)(116,242)
(117,243)(118,245)(119,244)(120,230)(121,229)(122,228)(123,232)(124,231)
(125,233)(126,234)(127,236)(128,235)(129,221)(130,220)(131,219)(132,223)
(133,222)(134,224)(135,225)(136,227)(137,226)(138,212)(139,211)(140,210)
(141,214)(142,213)(143,215)(144,216)(145,218)(146,217)(147,203)(148,202)
(149,201)(150,205)(151,204)(152,206)(153,207)(154,209)(155,208)(156,194)
(157,193)(158,192)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199);;
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)( 57,219)( 58,221)
( 59,220)( 60,223)( 61,222)( 62,224)( 63,227)( 64,226)( 65,225)( 66,237)
( 67,239)( 68,238)( 69,241)( 70,240)( 71,242)( 72,245)( 73,244)( 74,243)
( 75,228)( 76,230)( 77,229)( 78,232)( 79,231)( 80,233)( 81,236)( 82,235)
( 83,234)( 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)(139,140)(141,142)(144,146)(147,156)
(148,158)(149,157)(150,160)(151,159)(152,161)(153,164)(154,163)(155,162);;
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, 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);;
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*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,
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*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, 59)( 31, 58)( 32, 57)
( 33, 61)( 34, 60)( 35, 62)( 36, 63)( 37, 65)( 38, 64)( 39, 77)( 40, 76)
( 41, 75)( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)
( 49, 67)( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)
( 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)(111,239)(112,238)(113,237)(114,241)(115,240)
(116,242)(117,243)(118,245)(119,244)(120,230)(121,229)(122,228)(123,232)
(124,231)(125,233)(126,234)(127,236)(128,235)(129,221)(130,220)(131,219)
(132,223)(133,222)(134,224)(135,225)(136,227)(137,226)(138,212)(139,211)
(140,210)(141,214)(142,213)(143,215)(144,216)(145,218)(146,217)(147,203)
(148,202)(149,201)(150,205)(151,204)(152,206)(153,207)(154,209)(155,208)
(156,194)(157,193)(158,192)(159,196)(160,195)(161,197)(162,198)(163,200)
(164,199);
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)( 57,219)
( 58,221)( 59,220)( 60,223)( 61,222)( 62,224)( 63,227)( 64,226)( 65,225)
( 66,237)( 67,239)( 68,238)( 69,241)( 70,240)( 71,242)( 72,245)( 73,244)
( 74,243)( 75,228)( 76,230)( 77,229)( 78,232)( 79,231)( 80,233)( 81,236)
( 82,235)( 83,234)( 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)(139,140)(141,142)(144,146)
(147,156)(148,158)(149,157)(150,160)(151,159)(152,161)(153,164)(154,163)
(155,162);
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, 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);
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*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,
s3*s1*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2 >;
to this polytope