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