Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944e
if this polytope has a name.
Group : SmallGroup(1944,946)
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
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, 59)( 31, 58)( 32, 57)( 33, 65)
( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 73)( 40, 72)( 41, 74)
( 42, 70)( 43, 69)( 44, 71)( 45, 67)( 46, 66)( 47, 68)( 48, 78)( 49, 80)
( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 81)( 55, 83)( 56, 82)( 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)(111,224)(112,223)(113,222)(114,221)(115,220)(116,219)
(117,227)(118,226)(119,225)(120,229)(121,228)(122,230)(123,235)(124,234)
(125,236)(126,232)(127,231)(128,233)(129,243)(130,245)(131,244)(132,240)
(133,242)(134,241)(135,237)(136,239)(137,238)(138,197)(139,196)(140,195)
(141,194)(142,193)(143,192)(144,200)(145,199)(146,198)(147,202)(148,201)
(149,203)(150,208)(151,207)(152,209)(153,205)(154,204)(155,206)(156,216)
(157,218)(158,217)(159,213)(160,215)(161,214)(162,210)(163,212)(164,211);;
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)( 57,225)( 58,227)
( 59,226)( 60,222)( 61,224)( 62,223)( 63,219)( 64,221)( 65,220)( 66,235)
( 67,234)( 68,236)( 69,232)( 70,231)( 71,233)( 72,229)( 73,228)( 74,230)
( 75,245)( 76,244)( 77,243)( 78,242)( 79,241)( 80,240)( 81,239)( 82,238)
( 83,237)( 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)(138,144)(139,146)(140,145)(142,143)
(147,154)(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)(159,161);;
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, 59)( 31, 58)( 32, 57)( 33, 62)
( 34, 61)( 35, 60)( 36, 65)( 37, 64)( 38, 63)( 39, 80)( 40, 79)( 41, 78)
( 42, 83)( 43, 82)( 44, 81)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)
( 50, 72)( 51, 68)( 52, 67)( 53, 66)( 54, 71)( 55, 70)( 56, 69)( 85, 86)
( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)( 98,109)
( 99,102)(100,104)(101,103)(111,140)(112,139)(113,138)(114,143)(115,142)
(116,141)(117,146)(118,145)(119,144)(120,161)(121,160)(122,159)(123,164)
(124,163)(125,162)(126,158)(127,157)(128,156)(129,155)(130,154)(131,153)
(132,149)(133,148)(134,147)(135,152)(136,151)(137,150)(166,167)(169,170)
(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)(180,183)
(181,185)(182,184)(192,221)(193,220)(194,219)(195,224)(196,223)(197,222)
(198,227)(199,226)(200,225)(201,242)(202,241)(203,240)(204,245)(205,244)
(206,243)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)(213,230)
(214,229)(215,228)(216,233)(217,232)(218,231);;
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,
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,
s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*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)( 12, 20)( 13, 19)( 14, 18)
( 15, 17)( 21, 25)( 22, 24)( 23, 26)( 27, 28)( 30, 59)( 31, 58)( 32, 57)
( 33, 65)( 34, 64)( 35, 63)( 36, 62)( 37, 61)( 38, 60)( 39, 73)( 40, 72)
( 41, 74)( 42, 70)( 43, 69)( 44, 71)( 45, 67)( 46, 66)( 47, 68)( 48, 78)
( 49, 80)( 50, 79)( 51, 75)( 52, 77)( 53, 76)( 54, 81)( 55, 83)( 56, 82)
( 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)(111,224)(112,223)(113,222)(114,221)(115,220)
(116,219)(117,227)(118,226)(119,225)(120,229)(121,228)(122,230)(123,235)
(124,234)(125,236)(126,232)(127,231)(128,233)(129,243)(130,245)(131,244)
(132,240)(133,242)(134,241)(135,237)(136,239)(137,238)(138,197)(139,196)
(140,195)(141,194)(142,193)(143,192)(144,200)(145,199)(146,198)(147,202)
(148,201)(149,203)(150,208)(151,207)(152,209)(153,205)(154,204)(155,206)
(156,216)(157,218)(158,217)(159,213)(160,215)(161,214)(162,210)(163,212)
(164,211);
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)( 57,225)
( 58,227)( 59,226)( 60,222)( 61,224)( 62,223)( 63,219)( 64,221)( 65,220)
( 66,235)( 67,234)( 68,236)( 69,232)( 70,231)( 71,233)( 72,229)( 73,228)
( 74,230)( 75,245)( 76,244)( 77,243)( 78,242)( 79,241)( 80,240)( 81,239)
( 82,238)( 83,237)( 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)(138,144)(139,146)(140,145)
(142,143)(147,154)(148,153)(149,155)(150,151)(156,164)(157,163)(158,162)
(159,161);
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, 59)( 31, 58)( 32, 57)
( 33, 62)( 34, 61)( 35, 60)( 36, 65)( 37, 64)( 38, 63)( 39, 80)( 40, 79)
( 41, 78)( 42, 83)( 43, 82)( 44, 81)( 45, 77)( 46, 76)( 47, 75)( 48, 74)
( 49, 73)( 50, 72)( 51, 68)( 52, 67)( 53, 66)( 54, 71)( 55, 70)( 56, 69)
( 85, 86)( 88, 89)( 91, 92)( 93,105)( 94,107)( 95,106)( 96,108)( 97,110)
( 98,109)( 99,102)(100,104)(101,103)(111,140)(112,139)(113,138)(114,143)
(115,142)(116,141)(117,146)(118,145)(119,144)(120,161)(121,160)(122,159)
(123,164)(124,163)(125,162)(126,158)(127,157)(128,156)(129,155)(130,154)
(131,153)(132,149)(133,148)(134,147)(135,152)(136,151)(137,150)(166,167)
(169,170)(172,173)(174,186)(175,188)(176,187)(177,189)(178,191)(179,190)
(180,183)(181,185)(182,184)(192,221)(193,220)(194,219)(195,224)(196,223)
(197,222)(198,227)(199,226)(200,225)(201,242)(202,241)(203,240)(204,245)
(205,244)(206,243)(207,239)(208,238)(209,237)(210,236)(211,235)(212,234)
(213,230)(214,229)(215,228)(216,233)(217,232)(218,231);
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,
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,
s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2 >;
to this polytope