Polytope of Type {2,18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,18}*1944d
if this polytope has a name.
Group : SmallGroup(1944,945)
Rank : 4
Schlafli Type : {2,18,18}
Number of vertices, edges, etc : 2, 27, 243, 27
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Non-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 : {2,6,18}*648b, {2,18,6}*648c
9-fold quotients : {2,6,6}*216
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 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);;
s2 := ( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)( 10, 35)
( 11, 34)( 12, 41)( 13, 40)( 14, 39)( 15, 47)( 16, 46)( 17, 45)( 18, 44)
( 19, 43)( 20, 42)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)( 26, 56)
( 27, 52)( 28, 51)( 29, 53)( 57, 58)( 60, 64)( 61, 63)( 62, 65)( 67, 68)
( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)( 84,195)
( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)( 91,200)( 92,199)
( 93,206)( 94,205)( 95,204)( 96,203)( 97,202)( 98,201)( 99,209)(100,208)
(101,207)(102,214)(103,213)(104,215)(105,211)(106,210)(107,212)(108,217)
(109,216)(110,218)(111,168)(112,170)(113,169)(114,165)(115,167)(116,166)
(117,171)(118,173)(119,172)(120,179)(121,178)(122,177)(123,176)(124,175)
(125,174)(126,182)(127,181)(128,180)(129,187)(130,186)(131,188)(132,184)
(133,183)(134,185)(135,190)(136,189)(137,191)(138,223)(139,222)(140,224)
(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)(148,233)
(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)(156,242)
(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)(164,243);;
s3 := ( 3, 84)( 4, 85)( 5, 86)( 6, 90)( 7, 91)( 8, 92)( 9, 87)( 10, 88)
( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)( 18,105)
( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)( 26,101)
( 27, 96)( 28, 97)( 29, 98)( 30,120)( 31,121)( 32,122)( 33,126)( 34,127)
( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)( 42,117)
( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)( 50,131)
( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,159)( 58,160)
( 59,161)( 60,156)( 61,157)( 62,158)( 63,162)( 64,163)( 65,164)( 66,150)
( 67,151)( 68,152)( 69,147)( 70,148)( 71,149)( 72,153)( 73,154)( 74,155)
( 75,141)( 76,142)( 77,143)( 78,138)( 79,139)( 80,140)( 81,144)( 82,145)
( 83,146)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)(177,183)
(178,184)(179,185)(180,189)(181,190)(182,191)(192,204)(193,205)(194,206)
(195,201)(196,202)(197,203)(198,207)(199,208)(200,209)(210,213)(211,214)
(212,215)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)(225,237)
(226,238)(227,239)(228,234)(229,235)(230,236);;
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*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1,
s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := 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);
s2 := Sym(245)!( 3, 30)( 4, 32)( 5, 31)( 6, 36)( 7, 38)( 8, 37)( 9, 33)
( 10, 35)( 11, 34)( 12, 41)( 13, 40)( 14, 39)( 15, 47)( 16, 46)( 17, 45)
( 18, 44)( 19, 43)( 20, 42)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)
( 26, 56)( 27, 52)( 28, 51)( 29, 53)( 57, 58)( 60, 64)( 61, 63)( 62, 65)
( 67, 68)( 69, 72)( 70, 74)( 71, 73)( 75, 77)( 78, 83)( 79, 82)( 80, 81)
( 84,195)( 85,197)( 86,196)( 87,192)( 88,194)( 89,193)( 90,198)( 91,200)
( 92,199)( 93,206)( 94,205)( 95,204)( 96,203)( 97,202)( 98,201)( 99,209)
(100,208)(101,207)(102,214)(103,213)(104,215)(105,211)(106,210)(107,212)
(108,217)(109,216)(110,218)(111,168)(112,170)(113,169)(114,165)(115,167)
(116,166)(117,171)(118,173)(119,172)(120,179)(121,178)(122,177)(123,176)
(124,175)(125,174)(126,182)(127,181)(128,180)(129,187)(130,186)(131,188)
(132,184)(133,183)(134,185)(135,190)(136,189)(137,191)(138,223)(139,222)
(140,224)(141,220)(142,219)(143,221)(144,226)(145,225)(146,227)(147,231)
(148,233)(149,232)(150,228)(151,230)(152,229)(153,234)(154,236)(155,235)
(156,242)(157,241)(158,240)(159,239)(160,238)(161,237)(162,245)(163,244)
(164,243);
s3 := Sym(245)!( 3, 84)( 4, 85)( 5, 86)( 6, 90)( 7, 91)( 8, 92)( 9, 87)
( 10, 88)( 11, 89)( 12,102)( 13,103)( 14,104)( 15,108)( 16,109)( 17,110)
( 18,105)( 19,106)( 20,107)( 21, 93)( 22, 94)( 23, 95)( 24, 99)( 25,100)
( 26,101)( 27, 96)( 28, 97)( 29, 98)( 30,120)( 31,121)( 32,122)( 33,126)
( 34,127)( 35,128)( 36,123)( 37,124)( 38,125)( 39,111)( 40,112)( 41,113)
( 42,117)( 43,118)( 44,119)( 45,114)( 46,115)( 47,116)( 48,129)( 49,130)
( 50,131)( 51,135)( 52,136)( 53,137)( 54,132)( 55,133)( 56,134)( 57,159)
( 58,160)( 59,161)( 60,156)( 61,157)( 62,158)( 63,162)( 64,163)( 65,164)
( 66,150)( 67,151)( 68,152)( 69,147)( 70,148)( 71,149)( 72,153)( 73,154)
( 74,155)( 75,141)( 76,142)( 77,143)( 78,138)( 79,139)( 80,140)( 81,144)
( 82,145)( 83,146)(165,168)(166,169)(167,170)(174,186)(175,187)(176,188)
(177,183)(178,184)(179,185)(180,189)(181,190)(182,191)(192,204)(193,205)
(194,206)(195,201)(196,202)(197,203)(198,207)(199,208)(200,209)(210,213)
(211,214)(212,215)(219,243)(220,244)(221,245)(222,240)(223,241)(224,242)
(225,237)(226,238)(227,239)(228,234)(229,235)(230,236);
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*s2*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s1*s2*s1,
s1*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
to this polytope