Polytope of Type {18,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6,9}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2339)
Rank : 4
Schlafli Type : {18,6,9}
Number of vertices, edges, etc : 18, 54, 27, 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 : {18,2,9}*648, {6,6,9}*648b, {18,6,3}*648b
6-fold quotients : {9,2,9}*324
9-fold quotients : {2,6,9}*216, {6,2,9}*216, {18,2,3}*216, {6,6,3}*216b
18-fold quotients : {3,2,9}*108, {9,2,3}*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 := ( 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, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)( 8, 14)
( 9, 15)( 19, 22)( 20, 23)( 21, 24)( 28, 64)( 29, 65)( 30, 66)( 31, 70)
( 32, 71)( 33, 72)( 34, 67)( 35, 68)( 36, 69)( 37, 55)( 38, 56)( 39, 57)
( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 46, 76)( 47, 77)
( 48, 78)( 49, 73)( 50, 74)( 51, 75)( 52, 79)( 53, 80)( 54, 81)( 82, 91)
( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)
(100,103)(101,104)(102,105)(109,145)(110,146)(111,147)(112,151)(113,152)
(114,153)(115,148)(116,149)(117,150)(118,136)(119,137)(120,138)(121,142)
(122,143)(123,144)(124,139)(125,140)(126,141)(127,157)(128,158)(129,159)
(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)(163,172)(164,173)
(165,174)(166,178)(167,179)(168,180)(169,175)(170,176)(171,177)(181,184)
(182,185)(183,186)(190,226)(191,227)(192,228)(193,232)(194,233)(195,234)
(196,229)(197,230)(198,231)(199,217)(200,218)(201,219)(202,223)(203,224)
(204,225)(205,220)(206,221)(207,222)(208,238)(209,239)(210,240)(211,235)
(212,236)(213,237)(214,241)(215,242)(216,243);;
s2 := ( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)( 8, 36)
( 9, 35)( 10, 37)( 11, 39)( 12, 38)( 13, 40)( 14, 42)( 15, 41)( 16, 43)
( 17, 45)( 18, 44)( 19, 46)( 20, 48)( 21, 47)( 22, 49)( 23, 51)( 24, 50)
( 25, 52)( 26, 54)( 27, 53)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 82,191)( 83,190)( 84,192)( 85,194)
( 86,193)( 87,195)( 88,197)( 89,196)( 90,198)( 91,200)( 92,199)( 93,201)
( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)(100,209)(101,208)
(102,210)(103,212)(104,211)(105,213)(106,215)(107,214)(108,216)(109,164)
(110,163)(111,165)(112,167)(113,166)(114,168)(115,170)(116,169)(117,171)
(118,173)(119,172)(120,174)(121,176)(122,175)(123,177)(124,179)(125,178)
(126,180)(127,182)(128,181)(129,183)(130,185)(131,184)(132,186)(133,188)
(134,187)(135,189)(136,218)(137,217)(138,219)(139,221)(140,220)(141,222)
(142,224)(143,223)(144,225)(145,227)(146,226)(147,228)(148,230)(149,229)
(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)(156,237)(157,239)
(158,238)(159,240)(160,242)(161,241)(162,243);;
s3 := ( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)( 8, 90)
( 9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)( 16, 97)
( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)( 24,104)
( 25,106)( 26,108)( 27,107)( 28,136)( 29,138)( 30,137)( 31,139)( 32,141)
( 33,140)( 34,142)( 35,144)( 36,143)( 37,145)( 38,147)( 39,146)( 40,148)
( 41,150)( 42,149)( 43,151)( 44,153)( 45,152)( 46,154)( 47,156)( 48,155)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,109)( 56,111)
( 57,110)( 58,112)( 59,114)( 60,113)( 61,115)( 62,117)( 63,116)( 64,118)
( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)( 72,125)
( 73,127)( 74,129)( 75,128)( 76,130)( 77,132)( 78,131)( 79,133)( 80,135)
( 81,134)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)(181,182)
(184,185)(187,188)(190,218)(191,217)(192,219)(193,221)(194,220)(195,222)
(196,224)(197,223)(198,225)(199,227)(200,226)(201,228)(202,230)(203,229)
(204,231)(205,233)(206,232)(207,234)(208,236)(209,235)(210,237)(211,239)
(212,238)(213,240)(214,242)(215,241)(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, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
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, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)
( 8, 14)( 9, 15)( 19, 22)( 20, 23)( 21, 24)( 28, 64)( 29, 65)( 30, 66)
( 31, 70)( 32, 71)( 33, 72)( 34, 67)( 35, 68)( 36, 69)( 37, 55)( 38, 56)
( 39, 57)( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)( 46, 76)
( 47, 77)( 48, 78)( 49, 73)( 50, 74)( 51, 75)( 52, 79)( 53, 80)( 54, 81)
( 82, 91)( 83, 92)( 84, 93)( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)
( 90, 96)(100,103)(101,104)(102,105)(109,145)(110,146)(111,147)(112,151)
(113,152)(114,153)(115,148)(116,149)(117,150)(118,136)(119,137)(120,138)
(121,142)(122,143)(123,144)(124,139)(125,140)(126,141)(127,157)(128,158)
(129,159)(130,154)(131,155)(132,156)(133,160)(134,161)(135,162)(163,172)
(164,173)(165,174)(166,178)(167,179)(168,180)(169,175)(170,176)(171,177)
(181,184)(182,185)(183,186)(190,226)(191,227)(192,228)(193,232)(194,233)
(195,234)(196,229)(197,230)(198,231)(199,217)(200,218)(201,219)(202,223)
(203,224)(204,225)(205,220)(206,221)(207,222)(208,238)(209,239)(210,240)
(211,235)(212,236)(213,237)(214,241)(215,242)(216,243);
s2 := Sym(243)!( 1, 28)( 2, 30)( 3, 29)( 4, 31)( 5, 33)( 6, 32)( 7, 34)
( 8, 36)( 9, 35)( 10, 37)( 11, 39)( 12, 38)( 13, 40)( 14, 42)( 15, 41)
( 16, 43)( 17, 45)( 18, 44)( 19, 46)( 20, 48)( 21, 47)( 22, 49)( 23, 51)
( 24, 50)( 25, 52)( 26, 54)( 27, 53)( 56, 57)( 59, 60)( 62, 63)( 65, 66)
( 68, 69)( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 82,191)( 83,190)( 84,192)
( 85,194)( 86,193)( 87,195)( 88,197)( 89,196)( 90,198)( 91,200)( 92,199)
( 93,201)( 94,203)( 95,202)( 96,204)( 97,206)( 98,205)( 99,207)(100,209)
(101,208)(102,210)(103,212)(104,211)(105,213)(106,215)(107,214)(108,216)
(109,164)(110,163)(111,165)(112,167)(113,166)(114,168)(115,170)(116,169)
(117,171)(118,173)(119,172)(120,174)(121,176)(122,175)(123,177)(124,179)
(125,178)(126,180)(127,182)(128,181)(129,183)(130,185)(131,184)(132,186)
(133,188)(134,187)(135,189)(136,218)(137,217)(138,219)(139,221)(140,220)
(141,222)(142,224)(143,223)(144,225)(145,227)(146,226)(147,228)(148,230)
(149,229)(150,231)(151,233)(152,232)(153,234)(154,236)(155,235)(156,237)
(157,239)(158,238)(159,240)(160,242)(161,241)(162,243);
s3 := Sym(243)!( 1, 82)( 2, 84)( 3, 83)( 4, 85)( 5, 87)( 6, 86)( 7, 88)
( 8, 90)( 9, 89)( 10, 91)( 11, 93)( 12, 92)( 13, 94)( 14, 96)( 15, 95)
( 16, 97)( 17, 99)( 18, 98)( 19,100)( 20,102)( 21,101)( 22,103)( 23,105)
( 24,104)( 25,106)( 26,108)( 27,107)( 28,136)( 29,138)( 30,137)( 31,139)
( 32,141)( 33,140)( 34,142)( 35,144)( 36,143)( 37,145)( 38,147)( 39,146)
( 40,148)( 41,150)( 42,149)( 43,151)( 44,153)( 45,152)( 46,154)( 47,156)
( 48,155)( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,109)
( 56,111)( 57,110)( 58,112)( 59,114)( 60,113)( 61,115)( 62,117)( 63,116)
( 64,118)( 65,120)( 66,119)( 67,121)( 68,123)( 69,122)( 70,124)( 71,126)
( 72,125)( 73,127)( 74,129)( 75,128)( 76,130)( 77,132)( 78,131)( 79,133)
( 80,135)( 81,134)(163,164)(166,167)(169,170)(172,173)(175,176)(178,179)
(181,182)(184,185)(187,188)(190,218)(191,217)(192,219)(193,221)(194,220)
(195,222)(196,224)(197,223)(198,225)(199,227)(200,226)(201,228)(202,230)
(203,229)(204,231)(205,233)(206,232)(207,234)(208,236)(209,235)(210,237)
(211,239)(212,238)(213,240)(214,242)(215,241)(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,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope