Polytope of Type {9,6,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,6,3}*1944
if this polytope has a name.
Group : SmallGroup(1944,3577)
Rank : 5
Schlafli Type : {9,6,6,3}
Number of vertices, edges, etc : 9, 27, 18, 9, 3
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 : {9,2,6,3}*648, {9,6,2,3}*648, {3,6,6,3}*648
9-fold quotients : {9,2,2,3}*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 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 11, 12)( 13, 17)( 14, 16)( 15, 18)
( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)( 33, 36)
( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)( 51, 54)
( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)( 69, 72)
( 74, 75)( 76, 80)( 77, 79)( 78, 81)( 82,163)( 83,165)( 84,164)( 85,170)
( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,172)( 92,174)( 93,173)
( 94,179)( 95,178)( 96,180)( 97,176)( 98,175)( 99,177)(100,181)(101,183)
(102,182)(103,188)(104,187)(105,189)(106,185)(107,184)(108,186)(109,190)
(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)(117,195)
(118,199)(119,201)(120,200)(121,206)(122,205)(123,207)(124,203)(125,202)
(126,204)(127,208)(128,210)(129,209)(130,215)(131,214)(132,216)(133,212)
(134,211)(135,213)(136,217)(137,219)(138,218)(139,224)(140,223)(141,225)
(142,221)(143,220)(144,222)(145,226)(146,228)(147,227)(148,233)(149,232)
(150,234)(151,230)(152,229)(153,231)(154,235)(155,237)(156,236)(157,242)
(158,241)(159,243)(160,239)(161,238)(162,240);;
s1 := ( 1, 85)( 2, 87)( 3, 86)( 4, 82)( 5, 84)( 6, 83)( 7, 89)( 8, 88)
( 9, 90)( 10, 94)( 11, 96)( 12, 95)( 13, 91)( 14, 93)( 15, 92)( 16, 98)
( 17, 97)( 18, 99)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,107)( 26,106)( 27,108)( 28,112)( 29,114)( 30,113)( 31,109)( 32,111)
( 33,110)( 34,116)( 35,115)( 36,117)( 37,121)( 38,123)( 39,122)( 40,118)
( 41,120)( 42,119)( 43,125)( 44,124)( 45,126)( 46,130)( 47,132)( 48,131)
( 49,127)( 50,129)( 51,128)( 52,134)( 53,133)( 54,135)( 55,139)( 56,141)
( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)( 64,148)
( 65,150)( 66,149)( 67,145)( 68,147)( 69,146)( 70,152)( 71,151)( 72,153)
( 73,157)( 74,159)( 75,158)( 76,154)( 77,156)( 78,155)( 79,161)( 80,160)
( 81,162)(163,166)(164,168)(165,167)(169,170)(172,175)(173,177)(174,176)
(178,179)(181,184)(182,186)(183,185)(187,188)(190,193)(191,195)(192,194)
(196,197)(199,202)(200,204)(201,203)(205,206)(208,211)(209,213)(210,212)
(214,215)(217,220)(218,222)(219,221)(223,224)(226,229)(227,231)(228,230)
(232,233)(235,238)(236,240)(237,239)(241,242);;
s2 := ( 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);;
s3 := ( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)( 8, 35)
( 9, 36)( 10, 46)( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)( 16, 52)
( 17, 53)( 18, 54)( 19, 37)( 20, 38)( 21, 39)( 22, 40)( 23, 41)( 24, 42)
( 25, 43)( 26, 44)( 27, 45)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 82,109)( 83,110)( 84,111)( 85,112)
( 86,113)( 87,114)( 88,115)( 89,116)( 90,117)( 91,127)( 92,128)( 93,129)
( 94,130)( 95,131)( 96,132)( 97,133)( 98,134)( 99,135)(100,118)(101,119)
(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)(145,154)
(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207)(226,235)(227,236)(228,237)(229,238)(230,239)
(231,240)(232,241)(233,242)(234,243);;
s4 := ( 1, 10)( 2, 11)( 3, 12)( 4, 13)( 5, 14)( 6, 15)( 7, 16)( 8, 17)
( 9, 18)( 28, 64)( 29, 65)( 30, 66)( 31, 67)( 32, 68)( 33, 69)( 34, 70)
( 35, 71)( 36, 72)( 37, 55)( 38, 56)( 39, 57)( 40, 58)( 41, 59)( 42, 60)
( 43, 61)( 44, 62)( 45, 63)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)( 82, 91)( 83, 92)( 84, 93)( 85, 94)
( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)(111,147)
(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)(119,137)
(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)(127,154)
(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)
(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)
(171,180)(190,226)(191,227)(192,228)(193,229)(194,230)(195,231)(196,232)
(197,233)(198,234)(199,217)(200,218)(201,219)(202,220)(203,221)(204,222)
(205,223)(206,224)(207,225)(208,235)(209,236)(210,237)(211,238)(212,239)
(213,240)(214,241)(215,242)(216,243);;
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, s3*s4*s3*s4*s3*s4,
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, s0*s1*s2*s1*s0*s1*s0*s1*s2*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)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 11, 12)( 13, 17)( 14, 16)
( 15, 18)( 20, 21)( 22, 26)( 23, 25)( 24, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 38, 39)( 40, 44)( 41, 43)( 42, 45)( 47, 48)( 49, 53)( 50, 52)
( 51, 54)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 65, 66)( 67, 71)( 68, 70)
( 69, 72)( 74, 75)( 76, 80)( 77, 79)( 78, 81)( 82,163)( 83,165)( 84,164)
( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,172)( 92,174)
( 93,173)( 94,179)( 95,178)( 96,180)( 97,176)( 98,175)( 99,177)(100,181)
(101,183)(102,182)(103,188)(104,187)(105,189)(106,185)(107,184)(108,186)
(109,190)(110,192)(111,191)(112,197)(113,196)(114,198)(115,194)(116,193)
(117,195)(118,199)(119,201)(120,200)(121,206)(122,205)(123,207)(124,203)
(125,202)(126,204)(127,208)(128,210)(129,209)(130,215)(131,214)(132,216)
(133,212)(134,211)(135,213)(136,217)(137,219)(138,218)(139,224)(140,223)
(141,225)(142,221)(143,220)(144,222)(145,226)(146,228)(147,227)(148,233)
(149,232)(150,234)(151,230)(152,229)(153,231)(154,235)(155,237)(156,236)
(157,242)(158,241)(159,243)(160,239)(161,238)(162,240);
s1 := Sym(243)!( 1, 85)( 2, 87)( 3, 86)( 4, 82)( 5, 84)( 6, 83)( 7, 89)
( 8, 88)( 9, 90)( 10, 94)( 11, 96)( 12, 95)( 13, 91)( 14, 93)( 15, 92)
( 16, 98)( 17, 97)( 18, 99)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)
( 24,101)( 25,107)( 26,106)( 27,108)( 28,112)( 29,114)( 30,113)( 31,109)
( 32,111)( 33,110)( 34,116)( 35,115)( 36,117)( 37,121)( 38,123)( 39,122)
( 40,118)( 41,120)( 42,119)( 43,125)( 44,124)( 45,126)( 46,130)( 47,132)
( 48,131)( 49,127)( 50,129)( 51,128)( 52,134)( 53,133)( 54,135)( 55,139)
( 56,141)( 57,140)( 58,136)( 59,138)( 60,137)( 61,143)( 62,142)( 63,144)
( 64,148)( 65,150)( 66,149)( 67,145)( 68,147)( 69,146)( 70,152)( 71,151)
( 72,153)( 73,157)( 74,159)( 75,158)( 76,154)( 77,156)( 78,155)( 79,161)
( 80,160)( 81,162)(163,166)(164,168)(165,167)(169,170)(172,175)(173,177)
(174,176)(178,179)(181,184)(182,186)(183,185)(187,188)(190,193)(191,195)
(192,194)(196,197)(199,202)(200,204)(201,203)(205,206)(208,211)(209,213)
(210,212)(214,215)(217,220)(218,222)(219,221)(223,224)(226,229)(227,231)
(228,230)(232,233)(235,238)(236,240)(237,239)(241,242);
s2 := 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);
s3 := Sym(243)!( 1, 28)( 2, 29)( 3, 30)( 4, 31)( 5, 32)( 6, 33)( 7, 34)
( 8, 35)( 9, 36)( 10, 46)( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)
( 16, 52)( 17, 53)( 18, 54)( 19, 37)( 20, 38)( 21, 39)( 22, 40)( 23, 41)
( 24, 42)( 25, 43)( 26, 44)( 27, 45)( 64, 73)( 65, 74)( 66, 75)( 67, 76)
( 68, 77)( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 82,109)( 83,110)( 84,111)
( 85,112)( 86,113)( 87,114)( 88,115)( 89,116)( 90,117)( 91,127)( 92,128)
( 93,129)( 94,130)( 95,131)( 96,132)( 97,133)( 98,134)( 99,135)(100,118)
(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126)
(145,154)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)
(153,162)(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)
(170,197)(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)
(178,214)(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)
(186,204)(187,205)(188,206)(189,207)(226,235)(227,236)(228,237)(229,238)
(230,239)(231,240)(232,241)(233,242)(234,243);
s4 := Sym(243)!( 1, 10)( 2, 11)( 3, 12)( 4, 13)( 5, 14)( 6, 15)( 7, 16)
( 8, 17)( 9, 18)( 28, 64)( 29, 65)( 30, 66)( 31, 67)( 32, 68)( 33, 69)
( 34, 70)( 35, 71)( 36, 72)( 37, 55)( 38, 56)( 39, 57)( 40, 58)( 41, 59)
( 42, 60)( 43, 61)( 44, 62)( 45, 63)( 46, 73)( 47, 74)( 48, 75)( 49, 76)
( 50, 77)( 51, 78)( 52, 79)( 53, 80)( 54, 81)( 82, 91)( 83, 92)( 84, 93)
( 85, 94)( 86, 95)( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,145)(110,146)
(111,147)(112,148)(113,149)(114,150)(115,151)(116,152)(117,153)(118,136)
(119,137)(120,138)(121,139)(122,140)(123,141)(124,142)(125,143)(126,144)
(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)
(135,162)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)
(170,179)(171,180)(190,226)(191,227)(192,228)(193,229)(194,230)(195,231)
(196,232)(197,233)(198,234)(199,217)(200,218)(201,219)(202,220)(203,221)
(204,222)(205,223)(206,224)(207,225)(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,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, s3*s4*s3*s4*s3*s4, 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,
s0*s1*s2*s1*s0*s1*s0*s1*s2*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