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