Polytope of Type {2,18,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,18,18}*1944a
if this polytope has a name.
Group : SmallGroup(1944,940)
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}*648a, {2,18,6}*648a
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)( 6, 9)( 7, 11)( 8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 30, 63)( 31, 65)( 32, 64)( 33, 60)
( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)( 41, 73)
( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)( 49, 83)
( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)( 85, 86)
( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)
(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)(115,143)
(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)(123,150)
(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)(131,163)
(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)(168,171)
(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)(186,189)
(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)(197,223)
(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)(205,233)
(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)(213,240)
(214,242)(215,241)(216,237)(217,239)(218,238);;
s2 := ( 3, 30)( 4, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)( 10, 34)
( 11, 35)( 12, 48)( 13, 49)( 14, 50)( 15, 54)( 16, 55)( 17, 56)( 18, 51)
( 19, 52)( 20, 53)( 21, 39)( 22, 40)( 23, 41)( 24, 45)( 25, 46)( 26, 47)
( 27, 42)( 28, 43)( 29, 44)( 57, 63)( 58, 64)( 59, 65)( 66, 81)( 67, 82)
( 68, 83)( 69, 78)( 70, 79)( 71, 80)( 72, 75)( 73, 76)( 74, 77)( 84,210)
( 85,211)( 86,212)( 87,216)( 88,217)( 89,218)( 90,213)( 91,214)( 92,215)
( 93,201)( 94,202)( 95,203)( 96,207)( 97,208)( 98,209)( 99,204)(100,205)
(101,206)(102,192)(103,193)(104,194)(105,198)(106,199)(107,200)(108,195)
(109,196)(110,197)(111,183)(112,184)(113,185)(114,189)(115,190)(116,191)
(117,186)(118,187)(119,188)(120,174)(121,175)(122,176)(123,180)(124,181)
(125,182)(126,177)(127,178)(128,179)(129,165)(130,166)(131,167)(132,171)
(133,172)(134,173)(135,168)(136,169)(137,170)(138,243)(139,244)(140,245)
(141,240)(142,241)(143,242)(144,237)(145,238)(146,239)(147,234)(148,235)
(149,236)(150,231)(151,232)(152,233)(153,228)(154,229)(155,230)(156,225)
(157,226)(158,227)(159,222)(160,223)(161,224)(162,219)(163,220)(164,221);;
s3 := ( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)( 10, 92)
( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)( 18,108)
( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)( 26, 97)
( 27, 99)( 28,101)( 29,100)( 30,112)( 31,111)( 32,113)( 33,115)( 34,114)
( 35,116)( 36,118)( 37,117)( 38,119)( 39,130)( 40,129)( 41,131)( 42,133)
( 43,132)( 44,134)( 45,136)( 46,135)( 47,137)( 48,121)( 49,120)( 50,122)
( 51,124)( 52,123)( 53,125)( 54,127)( 55,126)( 56,128)( 57,140)( 58,139)
( 59,138)( 60,143)( 61,142)( 62,141)( 63,146)( 64,145)( 65,144)( 66,158)
( 67,157)( 68,156)( 69,161)( 70,160)( 71,159)( 72,164)( 73,163)( 74,162)
( 75,149)( 76,148)( 77,147)( 78,152)( 79,151)( 80,150)( 81,155)( 82,154)
( 83,153)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)(171,189)
(172,191)(173,190)(175,176)(178,179)(181,182)(192,211)(193,210)(194,212)
(195,214)(196,213)(197,215)(198,217)(199,216)(200,218)(201,202)(204,205)
(207,208)(219,239)(220,238)(221,237)(222,242)(223,241)(224,240)(225,245)
(226,244)(227,243)(228,230)(231,233)(234,236);;
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,
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2,
s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := 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, 63)( 31, 65)( 32, 64)
( 33, 60)( 34, 62)( 35, 61)( 36, 57)( 37, 59)( 38, 58)( 39, 72)( 40, 74)
( 41, 73)( 42, 69)( 43, 71)( 44, 70)( 45, 66)( 46, 68)( 47, 67)( 48, 81)
( 49, 83)( 50, 82)( 51, 78)( 52, 80)( 53, 79)( 54, 75)( 55, 77)( 56, 76)
( 85, 86)( 87, 90)( 88, 92)( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)
(103,104)(105,108)(106,110)(107,109)(111,144)(112,146)(113,145)(114,141)
(115,143)(116,142)(117,138)(118,140)(119,139)(120,153)(121,155)(122,154)
(123,150)(124,152)(125,151)(126,147)(127,149)(128,148)(129,162)(130,164)
(131,163)(132,159)(133,161)(134,160)(135,156)(136,158)(137,157)(166,167)
(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)(184,185)
(186,189)(187,191)(188,190)(192,225)(193,227)(194,226)(195,222)(196,224)
(197,223)(198,219)(199,221)(200,220)(201,234)(202,236)(203,235)(204,231)
(205,233)(206,232)(207,228)(208,230)(209,229)(210,243)(211,245)(212,244)
(213,240)(214,242)(215,241)(216,237)(217,239)(218,238);
s2 := Sym(245)!( 3, 30)( 4, 31)( 5, 32)( 6, 36)( 7, 37)( 8, 38)( 9, 33)
( 10, 34)( 11, 35)( 12, 48)( 13, 49)( 14, 50)( 15, 54)( 16, 55)( 17, 56)
( 18, 51)( 19, 52)( 20, 53)( 21, 39)( 22, 40)( 23, 41)( 24, 45)( 25, 46)
( 26, 47)( 27, 42)( 28, 43)( 29, 44)( 57, 63)( 58, 64)( 59, 65)( 66, 81)
( 67, 82)( 68, 83)( 69, 78)( 70, 79)( 71, 80)( 72, 75)( 73, 76)( 74, 77)
( 84,210)( 85,211)( 86,212)( 87,216)( 88,217)( 89,218)( 90,213)( 91,214)
( 92,215)( 93,201)( 94,202)( 95,203)( 96,207)( 97,208)( 98,209)( 99,204)
(100,205)(101,206)(102,192)(103,193)(104,194)(105,198)(106,199)(107,200)
(108,195)(109,196)(110,197)(111,183)(112,184)(113,185)(114,189)(115,190)
(116,191)(117,186)(118,187)(119,188)(120,174)(121,175)(122,176)(123,180)
(124,181)(125,182)(126,177)(127,178)(128,179)(129,165)(130,166)(131,167)
(132,171)(133,172)(134,173)(135,168)(136,169)(137,170)(138,243)(139,244)
(140,245)(141,240)(142,241)(143,242)(144,237)(145,238)(146,239)(147,234)
(148,235)(149,236)(150,231)(151,232)(152,233)(153,228)(154,229)(155,230)
(156,225)(157,226)(158,227)(159,222)(160,223)(161,224)(162,219)(163,220)
(164,221);
s3 := Sym(245)!( 3, 84)( 4, 86)( 5, 85)( 6, 87)( 7, 89)( 8, 88)( 9, 90)
( 10, 92)( 11, 91)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)( 17,106)
( 18,108)( 19,110)( 20,109)( 21, 93)( 22, 95)( 23, 94)( 24, 96)( 25, 98)
( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,112)( 31,111)( 32,113)( 33,115)
( 34,114)( 35,116)( 36,118)( 37,117)( 38,119)( 39,130)( 40,129)( 41,131)
( 42,133)( 43,132)( 44,134)( 45,136)( 46,135)( 47,137)( 48,121)( 49,120)
( 50,122)( 51,124)( 52,123)( 53,125)( 54,127)( 55,126)( 56,128)( 57,140)
( 58,139)( 59,138)( 60,143)( 61,142)( 62,141)( 63,146)( 64,145)( 65,144)
( 66,158)( 67,157)( 68,156)( 69,161)( 70,160)( 71,159)( 72,164)( 73,163)
( 74,162)( 75,149)( 76,148)( 77,147)( 78,152)( 79,151)( 80,150)( 81,155)
( 82,154)( 83,153)(165,183)(166,185)(167,184)(168,186)(169,188)(170,187)
(171,189)(172,191)(173,190)(175,176)(178,179)(181,182)(192,211)(193,210)
(194,212)(195,214)(196,213)(197,215)(198,217)(199,216)(200,218)(201,202)
(204,205)(207,208)(219,239)(220,238)(221,237)(222,242)(223,241)(224,240)
(225,245)(226,244)(227,243)(228,230)(231,233)(234,236);
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, s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*s2,
s3*s2*s3*s1*s2*s1*s2*s3*s2*s3*s2*s3*s2*s1*s2*s3*s1*s2,
s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3 >;
to this polytope