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