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