Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944d
if this polytope has a name.
Group : SmallGroup(1944,945)
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}*648d, {2,3,18}*648
9-fold quotients : {2,3,6}*216
27-fold quotients : {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, 9)( 7, 11)( 8, 10)( 12, 20)( 13, 19)( 14, 18)( 15, 17)
( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 58)( 31, 57)( 32, 59)( 33, 64)
( 34, 63)( 35, 65)( 36, 61)( 37, 60)( 38, 62)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 80)( 49, 79)
( 50, 78)( 51, 77)( 52, 76)( 53, 75)( 54, 83)( 55, 82)( 56, 81)( 84,168)
( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)( 90,171)( 91,173)( 92,172)
( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)(100,178)
(101,177)(102,190)(103,189)(104,191)(105,187)(106,186)(107,188)(108,184)
(109,183)(110,185)(111,223)(112,222)(113,224)(114,220)(115,219)(116,221)
(117,226)(118,225)(119,227)(120,228)(121,230)(122,229)(123,234)(124,236)
(125,235)(126,231)(127,233)(128,232)(129,245)(130,244)(131,243)(132,242)
(133,241)(134,240)(135,239)(136,238)(137,237)(138,196)(139,195)(140,197)
(141,193)(142,192)(143,194)(144,199)(145,198)(146,200)(147,201)(148,203)
(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)(155,205)(156,218)
(157,217)(158,216)(159,215)(160,214)(161,213)(162,212)(163,211)(164,210);;
s2 := ( 3,201)( 4,203)( 5,202)( 6,207)( 7,209)( 8,208)( 9,204)( 10,206)
( 11,205)( 12,211)( 13,210)( 14,212)( 15,217)( 16,216)( 17,218)( 18,214)
( 19,213)( 20,215)( 21,194)( 22,193)( 23,192)( 24,200)( 25,199)( 26,198)
( 27,197)( 28,196)( 29,195)( 30,185)( 31,184)( 32,183)( 33,191)( 34,190)
( 35,189)( 36,188)( 37,187)( 38,186)( 39,165)( 40,167)( 41,166)( 42,171)
( 43,173)( 44,172)( 45,168)( 46,170)( 47,169)( 48,175)( 49,174)( 50,176)
( 51,181)( 52,180)( 53,182)( 54,178)( 55,177)( 56,179)( 57,227)( 58,226)
( 59,225)( 60,224)( 61,223)( 62,222)( 63,221)( 64,220)( 65,219)( 66,234)
( 67,236)( 68,235)( 69,231)( 70,233)( 71,232)( 72,228)( 73,230)( 74,229)
( 75,244)( 76,243)( 77,245)( 78,241)( 79,240)( 80,242)( 81,238)( 82,237)
( 83,239)( 84,120)( 85,122)( 86,121)( 87,126)( 88,128)( 89,127)( 90,123)
( 91,125)( 92,124)( 93,130)( 94,129)( 95,131)( 96,136)( 97,135)( 98,137)
( 99,133)(100,132)(101,134)(102,113)(103,112)(104,111)(105,119)(106,118)
(107,117)(108,116)(109,115)(110,114)(138,146)(139,145)(140,144)(141,143)
(147,153)(148,155)(149,154)(151,152)(156,163)(157,162)(158,164)(159,160);;
s3 := ( 4, 5)( 7, 8)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 27)( 16, 29)
( 17, 28)( 18, 21)( 19, 23)( 20, 22)( 30, 58)( 31, 57)( 32, 59)( 33, 61)
( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)( 41, 80)
( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)( 49, 72)
( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)( 85, 86)
( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)
( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)(115,141)
(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)(123,163)
(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)(131,155)
(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)(169,170)
(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)
(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)(197,224)
(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)(205,243)
(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)(213,229)
(214,228)(215,230)(216,232)(217,231)(218,233);;
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,
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1,
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, 9)( 7, 11)( 8, 10)( 12, 20)( 13, 19)( 14, 18)
( 15, 17)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 58)( 31, 57)( 32, 59)
( 33, 64)( 34, 63)( 35, 65)( 36, 61)( 37, 60)( 38, 62)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 80)
( 49, 79)( 50, 78)( 51, 77)( 52, 76)( 53, 75)( 54, 83)( 55, 82)( 56, 81)
( 84,168)( 85,170)( 86,169)( 87,165)( 88,167)( 89,166)( 90,171)( 91,173)
( 92,172)( 93,176)( 94,175)( 95,174)( 96,182)( 97,181)( 98,180)( 99,179)
(100,178)(101,177)(102,190)(103,189)(104,191)(105,187)(106,186)(107,188)
(108,184)(109,183)(110,185)(111,223)(112,222)(113,224)(114,220)(115,219)
(116,221)(117,226)(118,225)(119,227)(120,228)(121,230)(122,229)(123,234)
(124,236)(125,235)(126,231)(127,233)(128,232)(129,245)(130,244)(131,243)
(132,242)(133,241)(134,240)(135,239)(136,238)(137,237)(138,196)(139,195)
(140,197)(141,193)(142,192)(143,194)(144,199)(145,198)(146,200)(147,201)
(148,203)(149,202)(150,207)(151,209)(152,208)(153,204)(154,206)(155,205)
(156,218)(157,217)(158,216)(159,215)(160,214)(161,213)(162,212)(163,211)
(164,210);
s2 := Sym(245)!( 3,201)( 4,203)( 5,202)( 6,207)( 7,209)( 8,208)( 9,204)
( 10,206)( 11,205)( 12,211)( 13,210)( 14,212)( 15,217)( 16,216)( 17,218)
( 18,214)( 19,213)( 20,215)( 21,194)( 22,193)( 23,192)( 24,200)( 25,199)
( 26,198)( 27,197)( 28,196)( 29,195)( 30,185)( 31,184)( 32,183)( 33,191)
( 34,190)( 35,189)( 36,188)( 37,187)( 38,186)( 39,165)( 40,167)( 41,166)
( 42,171)( 43,173)( 44,172)( 45,168)( 46,170)( 47,169)( 48,175)( 49,174)
( 50,176)( 51,181)( 52,180)( 53,182)( 54,178)( 55,177)( 56,179)( 57,227)
( 58,226)( 59,225)( 60,224)( 61,223)( 62,222)( 63,221)( 64,220)( 65,219)
( 66,234)( 67,236)( 68,235)( 69,231)( 70,233)( 71,232)( 72,228)( 73,230)
( 74,229)( 75,244)( 76,243)( 77,245)( 78,241)( 79,240)( 80,242)( 81,238)
( 82,237)( 83,239)( 84,120)( 85,122)( 86,121)( 87,126)( 88,128)( 89,127)
( 90,123)( 91,125)( 92,124)( 93,130)( 94,129)( 95,131)( 96,136)( 97,135)
( 98,137)( 99,133)(100,132)(101,134)(102,113)(103,112)(104,111)(105,119)
(106,118)(107,117)(108,116)(109,115)(110,114)(138,146)(139,145)(140,144)
(141,143)(147,153)(148,155)(149,154)(151,152)(156,163)(157,162)(158,164)
(159,160);
s3 := Sym(245)!( 4, 5)( 7, 8)( 10, 11)( 12, 24)( 13, 26)( 14, 25)( 15, 27)
( 16, 29)( 17, 28)( 18, 21)( 19, 23)( 20, 22)( 30, 58)( 31, 57)( 32, 59)
( 33, 61)( 34, 60)( 35, 62)( 36, 64)( 37, 63)( 38, 65)( 39, 79)( 40, 78)
( 41, 80)( 42, 82)( 43, 81)( 44, 83)( 45, 76)( 46, 75)( 47, 77)( 48, 73)
( 49, 72)( 50, 74)( 51, 67)( 52, 66)( 53, 68)( 54, 70)( 55, 69)( 56, 71)
( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)
( 98,109)( 99,102)(100,104)(101,103)(111,139)(112,138)(113,140)(114,142)
(115,141)(116,143)(117,145)(118,144)(119,146)(120,160)(121,159)(122,161)
(123,163)(124,162)(125,164)(126,157)(127,156)(128,158)(129,154)(130,153)
(131,155)(132,148)(133,147)(134,149)(135,151)(136,150)(137,152)(166,167)
(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)
(180,183)(181,185)(182,184)(192,220)(193,219)(194,221)(195,223)(196,222)
(197,224)(198,226)(199,225)(200,227)(201,241)(202,240)(203,242)(204,244)
(205,243)(206,245)(207,238)(208,237)(209,239)(210,235)(211,234)(212,236)
(213,229)(214,228)(215,230)(216,232)(217,231)(218,233);
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,
s2*s3*s2*s1*s3*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1,
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