Polytope of Type {6,18,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 4
Schlafli Type : {6,18,9}
Number of vertices, edges, etc : 6, 54, 81, 9
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 2
Special Properties :
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 : {2,18,9}*648, {6,6,9}*648b
9-fold quotients : {2,6,9}*216, {6,2,9}*216, {6,6,3}*216b
18-fold quotients : {3,2,9}*108
27-fold quotients : {2,2,9}*72, {2,6,3}*72, {6,2,3}*72
54-fold quotients : {3,2,3}*36
81-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)
(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)
(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)
(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(190,217)(191,218)
(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)
(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)
(208,235)(209,236)(210,237)(211,238)(212,239)(213,240)(214,241)(215,242)
(216,243);;
s1 := ( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)( 8, 32)
( 9, 33)( 10, 49)( 11, 50)( 12, 51)( 13, 46)( 14, 47)( 15, 48)( 16, 52)
( 17, 53)( 18, 54)( 19, 40)( 20, 41)( 21, 42)( 22, 37)( 23, 38)( 24, 39)
( 25, 43)( 26, 44)( 27, 45)( 58, 61)( 59, 62)( 60, 63)( 64, 76)( 65, 77)
( 66, 78)( 67, 73)( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82,109)
( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)( 90,114)
( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)( 97,133)( 98,134)
( 99,135)(100,121)(101,122)(102,123)(103,118)(104,119)(105,120)(106,124)
(107,125)(108,126)(139,142)(140,143)(141,144)(145,157)(146,158)(147,159)
(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)(163,190)(164,191)
(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)(172,211)
(173,212)(174,213)(175,208)(176,209)(177,210)(178,214)(179,215)(180,216)
(181,202)(182,203)(183,204)(184,199)(185,200)(186,201)(187,205)(188,206)
(189,207)(220,223)(221,224)(222,225)(226,238)(227,239)(228,240)(229,235)
(230,236)(231,237)(232,241)(233,242)(234,243);;
s2 := ( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)( 8, 15)
( 9, 14)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 49)( 47, 51)
( 48, 50)( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82,173)
( 83,172)( 84,174)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)( 90,177)
( 91,164)( 92,163)( 93,165)( 94,170)( 95,169)( 96,171)( 97,167)( 98,166)
( 99,168)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)(106,188)
(107,187)(108,189)(109,200)(110,199)(111,201)(112,206)(113,205)(114,207)
(115,203)(116,202)(117,204)(118,191)(119,190)(120,192)(121,197)(122,196)
(123,198)(124,194)(125,193)(126,195)(127,212)(128,211)(129,213)(130,209)
(131,208)(132,210)(133,215)(134,214)(135,216)(136,227)(137,226)(138,228)
(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,218)(146,217)
(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)(154,239)
(155,238)(156,240)(157,236)(158,235)(159,237)(160,242)(161,241)(162,243);;
s3 := ( 1, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)( 8, 87)
( 9, 86)( 10,103)( 11,105)( 12,104)( 13,100)( 14,102)( 15,101)( 16,106)
( 17,108)( 18,107)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)( 24, 92)
( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,115)( 32,117)
( 33,116)( 34,112)( 35,114)( 36,113)( 37,130)( 38,132)( 39,131)( 40,127)
( 41,129)( 42,128)( 43,133)( 44,135)( 45,134)( 46,121)( 47,123)( 48,122)
( 49,118)( 50,120)( 51,119)( 52,124)( 53,126)( 54,125)( 55,136)( 56,138)
( 57,137)( 58,142)( 59,144)( 60,143)( 61,139)( 62,141)( 63,140)( 64,157)
( 65,159)( 66,158)( 67,154)( 68,156)( 69,155)( 70,160)( 71,162)( 72,161)
( 73,148)( 74,150)( 75,149)( 76,145)( 77,147)( 78,146)( 79,151)( 80,153)
( 81,152)(163,164)(166,170)(167,169)(168,171)(172,185)(173,184)(174,186)
(175,182)(176,181)(177,183)(178,188)(179,187)(180,189)(190,191)(193,197)
(194,196)(195,198)(199,212)(200,211)(201,213)(202,209)(203,208)(204,210)
(205,215)(206,214)(207,216)(217,218)(220,224)(221,223)(222,225)(226,239)
(227,238)(228,240)(229,236)(230,235)(231,237)(232,242)(233,241)(234,243);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)
(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(190,217)
(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)
(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)
(207,234)(208,235)(209,236)(210,237)(211,238)(212,239)(213,240)(214,241)
(215,242)(216,243);
s1 := Sym(243)!( 1, 28)( 2, 29)( 3, 30)( 4, 34)( 5, 35)( 6, 36)( 7, 31)
( 8, 32)( 9, 33)( 10, 49)( 11, 50)( 12, 51)( 13, 46)( 14, 47)( 15, 48)
( 16, 52)( 17, 53)( 18, 54)( 19, 40)( 20, 41)( 21, 42)( 22, 37)( 23, 38)
( 24, 39)( 25, 43)( 26, 44)( 27, 45)( 58, 61)( 59, 62)( 60, 63)( 64, 76)
( 65, 77)( 66, 78)( 67, 73)( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)
( 82,109)( 83,110)( 84,111)( 85,115)( 86,116)( 87,117)( 88,112)( 89,113)
( 90,114)( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)( 97,133)
( 98,134)( 99,135)(100,121)(101,122)(102,123)(103,118)(104,119)(105,120)
(106,124)(107,125)(108,126)(139,142)(140,143)(141,144)(145,157)(146,158)
(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,211)(173,212)(174,213)(175,208)(176,209)(177,210)(178,214)(179,215)
(180,216)(181,202)(182,203)(183,204)(184,199)(185,200)(186,201)(187,205)
(188,206)(189,207)(220,223)(221,224)(222,225)(226,238)(227,239)(228,240)
(229,235)(230,236)(231,237)(232,241)(233,242)(234,243);
s2 := Sym(243)!( 1, 10)( 2, 12)( 3, 11)( 4, 16)( 5, 18)( 6, 17)( 7, 13)
( 8, 15)( 9, 14)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 49)
( 47, 51)( 48, 50)( 53, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 76)( 74, 78)( 75, 77)( 80, 81)
( 82,173)( 83,172)( 84,174)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)
( 90,177)( 91,164)( 92,163)( 93,165)( 94,170)( 95,169)( 96,171)( 97,167)
( 98,166)( 99,168)(100,185)(101,184)(102,186)(103,182)(104,181)(105,183)
(106,188)(107,187)(108,189)(109,200)(110,199)(111,201)(112,206)(113,205)
(114,207)(115,203)(116,202)(117,204)(118,191)(119,190)(120,192)(121,197)
(122,196)(123,198)(124,194)(125,193)(126,195)(127,212)(128,211)(129,213)
(130,209)(131,208)(132,210)(133,215)(134,214)(135,216)(136,227)(137,226)
(138,228)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,218)
(146,217)(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)
(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,242)(161,241)
(162,243);
s3 := Sym(243)!( 1, 82)( 2, 84)( 3, 83)( 4, 88)( 5, 90)( 6, 89)( 7, 85)
( 8, 87)( 9, 86)( 10,103)( 11,105)( 12,104)( 13,100)( 14,102)( 15,101)
( 16,106)( 17,108)( 18,107)( 19, 94)( 20, 96)( 21, 95)( 22, 91)( 23, 93)
( 24, 92)( 25, 97)( 26, 99)( 27, 98)( 28,109)( 29,111)( 30,110)( 31,115)
( 32,117)( 33,116)( 34,112)( 35,114)( 36,113)( 37,130)( 38,132)( 39,131)
( 40,127)( 41,129)( 42,128)( 43,133)( 44,135)( 45,134)( 46,121)( 47,123)
( 48,122)( 49,118)( 50,120)( 51,119)( 52,124)( 53,126)( 54,125)( 55,136)
( 56,138)( 57,137)( 58,142)( 59,144)( 60,143)( 61,139)( 62,141)( 63,140)
( 64,157)( 65,159)( 66,158)( 67,154)( 68,156)( 69,155)( 70,160)( 71,162)
( 72,161)( 73,148)( 74,150)( 75,149)( 76,145)( 77,147)( 78,146)( 79,151)
( 80,153)( 81,152)(163,164)(166,170)(167,169)(168,171)(172,185)(173,184)
(174,186)(175,182)(176,181)(177,183)(178,188)(179,187)(180,189)(190,191)
(193,197)(194,196)(195,198)(199,212)(200,211)(201,213)(202,209)(203,208)
(204,210)(205,215)(206,214)(207,216)(217,218)(220,224)(221,223)(222,225)
(226,239)(227,238)(228,240)(229,236)(230,235)(231,237)(232,242)(233,241)
(234,243);
poly := sub<Sym(243)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope