Polytope of Type {3,6,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6,9}*1944
if this polytope has a name.
Group : SmallGroup(1944,3577)
Rank : 5
Schlafli Type : {3,6,6,9}
Number of vertices, edges, etc : 3, 9, 18, 27, 9
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 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 : {3,2,6,9}*648, {3,6,2,9}*648, {3,6,6,3}*648
9-fold quotients : {3,2,2,9}*216, {3,2,6,3}*216, {3,6,2,3}*216
27-fold quotients : {3,2,2,3}*72
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)( 82,163)( 83,164)( 84,165)( 85,166)( 86,167)
( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)( 94,175)
( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)(102,183)
(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,217)(110,218)
(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)(118,226)
(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)(126,234)
(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)(134,242)
(135,243)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216);;
s1 := ( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)( 8,116)
( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)( 16,124)
( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,133)( 26,134)( 27,135)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,136)( 56,137)
( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)( 64,145)
( 65,146)( 66,147)( 67,148)( 68,149)( 69,150)( 70,151)( 71,152)( 72,153)
( 73,154)( 74,155)( 75,156)( 76,157)( 77,158)( 78,159)( 79,160)( 80,161)
( 81,162)(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);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)( 42, 78)
( 43, 79)( 44, 80)( 45, 81)( 46, 64)( 47, 65)( 48, 66)( 49, 67)( 50, 68)
( 51, 69)( 52, 70)( 53, 71)( 54, 72)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)
(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)
(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)
(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)
(172,181)(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)(179,188)
(180,189)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)
(197,224)(198,225)(199,235)(200,236)(201,237)(202,238)(203,239)(204,240)
(205,241)(206,242)(207,243)(208,226)(209,227)(210,228)(211,229)(212,230)
(213,231)(214,232)(215,233)(216,234);;
s3 := ( 1, 10)( 2, 12)( 3, 11)( 4, 17)( 5, 16)( 6, 18)( 7, 14)( 8, 13)
( 9, 15)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 47, 48)( 49, 53)
( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)( 60, 72)
( 61, 68)( 62, 67)( 63, 69)( 74, 75)( 76, 80)( 77, 79)( 78, 81)( 82, 91)
( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)( 90, 96)
(101,102)(103,107)(104,106)(105,108)(109,118)(110,120)(111,119)(112,125)
(113,124)(114,126)(115,122)(116,121)(117,123)(128,129)(130,134)(131,133)
(132,135)(136,145)(137,147)(138,146)(139,152)(140,151)(141,153)(142,149)
(143,148)(144,150)(155,156)(157,161)(158,160)(159,162)(163,172)(164,174)
(165,173)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)(182,183)
(184,188)(185,187)(186,189)(190,199)(191,201)(192,200)(193,206)(194,205)
(195,207)(196,203)(197,202)(198,204)(209,210)(211,215)(212,214)(213,216)
(217,226)(218,228)(219,227)(220,233)(221,232)(222,234)(223,230)(224,229)
(225,231)(236,237)(238,242)(239,241)(240,243);;
s4 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 28, 31)( 29, 33)( 30, 32)
( 34, 35)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)
( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 62)( 64, 76)( 65, 78)
( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 80)( 71, 79)( 72, 81)( 82, 85)
( 83, 87)( 84, 86)( 88, 89)( 91,103)( 92,105)( 93,104)( 94,100)( 95,102)
( 96,101)( 97,107)( 98,106)( 99,108)(109,112)(110,114)(111,113)(115,116)
(118,130)(119,132)(120,131)(121,127)(122,129)(123,128)(124,134)(125,133)
(126,135)(136,139)(137,141)(138,140)(142,143)(145,157)(146,159)(147,158)
(148,154)(149,156)(150,155)(151,161)(152,160)(153,162)(163,166)(164,168)
(165,167)(169,170)(172,184)(173,186)(174,185)(175,181)(176,183)(177,182)
(178,188)(179,187)(180,189)(190,193)(191,195)(192,194)(196,197)(199,211)
(200,213)(201,212)(202,208)(203,210)(204,209)(205,215)(206,214)(207,216)
(217,220)(218,222)(219,221)(223,224)(226,238)(227,240)(228,239)(229,235)
(230,237)(231,236)(232,242)(233,241)(234,243);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
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)( 82,163)( 83,164)( 84,165)( 85,166)
( 86,167)( 87,168)( 88,169)( 89,170)( 90,171)( 91,172)( 92,173)( 93,174)
( 94,175)( 95,176)( 96,177)( 97,178)( 98,179)( 99,180)(100,181)(101,182)
(102,183)(103,184)(104,185)(105,186)(106,187)(107,188)(108,189)(109,217)
(110,218)(111,219)(112,220)(113,221)(114,222)(115,223)(116,224)(117,225)
(118,226)(119,227)(120,228)(121,229)(122,230)(123,231)(124,232)(125,233)
(126,234)(127,235)(128,236)(129,237)(130,238)(131,239)(132,240)(133,241)
(134,242)(135,243)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)
(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)
(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)
(158,212)(159,213)(160,214)(161,215)(162,216);
s1 := Sym(243)!( 1,109)( 2,110)( 3,111)( 4,112)( 5,113)( 6,114)( 7,115)
( 8,116)( 9,117)( 10,118)( 11,119)( 12,120)( 13,121)( 14,122)( 15,123)
( 16,124)( 17,125)( 18,126)( 19,127)( 20,128)( 21,129)( 22,130)( 23,131)
( 24,132)( 25,133)( 26,134)( 27,135)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)( 55,136)
( 56,137)( 57,138)( 58,139)( 59,140)( 60,141)( 61,142)( 62,143)( 63,144)
( 64,145)( 65,146)( 66,147)( 67,148)( 68,149)( 69,150)( 70,151)( 71,152)
( 72,153)( 73,154)( 74,155)( 75,156)( 76,157)( 77,158)( 78,159)( 79,160)
( 80,161)( 81,162)(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);
s2 := Sym(243)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)
( 34, 61)( 35, 62)( 36, 63)( 37, 73)( 38, 74)( 39, 75)( 40, 76)( 41, 77)
( 42, 78)( 43, 79)( 44, 80)( 45, 81)( 46, 64)( 47, 65)( 48, 66)( 49, 67)
( 50, 68)( 51, 69)( 52, 70)( 53, 71)( 54, 72)( 91,100)( 92,101)( 93,102)
( 94,103)( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)
(111,138)(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)
(119,155)(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)
(127,145)(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)
(135,153)(172,181)(173,182)(174,183)(175,184)(176,185)(177,186)(178,187)
(179,188)(180,189)(190,217)(191,218)(192,219)(193,220)(194,221)(195,222)
(196,223)(197,224)(198,225)(199,235)(200,236)(201,237)(202,238)(203,239)
(204,240)(205,241)(206,242)(207,243)(208,226)(209,227)(210,228)(211,229)
(212,230)(213,231)(214,232)(215,233)(216,234);
s3 := Sym(243)!( 1, 10)( 2, 12)( 3, 11)( 4, 17)( 5, 16)( 6, 18)( 7, 14)
( 8, 13)( 9, 15)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)
( 30, 38)( 31, 44)( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 47, 48)
( 49, 53)( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 74, 75)( 76, 80)( 77, 79)( 78, 81)
( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)
( 90, 96)(101,102)(103,107)(104,106)(105,108)(109,118)(110,120)(111,119)
(112,125)(113,124)(114,126)(115,122)(116,121)(117,123)(128,129)(130,134)
(131,133)(132,135)(136,145)(137,147)(138,146)(139,152)(140,151)(141,153)
(142,149)(143,148)(144,150)(155,156)(157,161)(158,160)(159,162)(163,172)
(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)(170,175)(171,177)
(182,183)(184,188)(185,187)(186,189)(190,199)(191,201)(192,200)(193,206)
(194,205)(195,207)(196,203)(197,202)(198,204)(209,210)(211,215)(212,214)
(213,216)(217,226)(218,228)(219,227)(220,233)(221,232)(222,234)(223,230)
(224,229)(225,231)(236,237)(238,242)(239,241)(240,243);
s4 := Sym(243)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)( 10, 22)( 11, 24)( 12, 23)
( 13, 19)( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 28, 31)( 29, 33)
( 30, 32)( 34, 35)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 53)( 44, 52)( 45, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 62)( 64, 76)
( 65, 78)( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 80)( 71, 79)( 72, 81)
( 82, 85)( 83, 87)( 84, 86)( 88, 89)( 91,103)( 92,105)( 93,104)( 94,100)
( 95,102)( 96,101)( 97,107)( 98,106)( 99,108)(109,112)(110,114)(111,113)
(115,116)(118,130)(119,132)(120,131)(121,127)(122,129)(123,128)(124,134)
(125,133)(126,135)(136,139)(137,141)(138,140)(142,143)(145,157)(146,159)
(147,158)(148,154)(149,156)(150,155)(151,161)(152,160)(153,162)(163,166)
(164,168)(165,167)(169,170)(172,184)(173,186)(174,185)(175,181)(176,183)
(177,182)(178,188)(179,187)(180,189)(190,193)(191,195)(192,194)(196,197)
(199,211)(200,213)(201,212)(202,208)(203,210)(204,209)(205,215)(206,214)
(207,216)(217,220)(218,222)(219,221)(223,224)(226,238)(227,240)(228,239)
(229,235)(230,237)(231,236)(232,242)(233,241)(234,243);
poly := sub<Sym(243)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;
References : None.
to this polytope