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Polytope of Type {6,6,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,27}*1944b
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 4
Schlafli Type : {6,6,27}
Number of vertices, edges, etc : 6, 18, 81, 27
Order of s0s1s2s3 : 54
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 : {2,6,27}*648, {6,2,27}*648, {6,6,9}*648b
   6-fold quotients : {3,2,27}*324
   9-fold quotients : {2,2,27}*216, {2,6,9}*216, {6,2,9}*216, {6,6,3}*216b
   18-fold quotients : {3,2,9}*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 := ( 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)(109,190)(110,191)(111,192)(112,193)(113,194)
(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)(121,202)
(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)(129,210)
(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)(137,218)
(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)(145,226)
(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)(153,234)
(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)(161,242)
(162,243);;
s1 := (  1, 82)(  2, 83)(  3, 84)(  4, 85)(  5, 86)(  6, 87)(  7, 88)(  8, 89)
(  9, 90)( 10, 91)( 11, 92)( 12, 93)( 13, 94)( 14, 95)( 15, 96)( 16, 97)
( 17, 98)( 18, 99)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)( 24,105)
( 25,106)( 26,107)( 27,108)( 28,136)( 29,137)( 30,138)( 31,139)( 32,140)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)( 40,148)
( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,109)( 56,110)
( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)( 64,118)
( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)( 72,126)
( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)( 80,134)
( 81,135)(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);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 35)(  5, 34)(  6, 36)(  7, 32)(  8, 31)
(  9, 33)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)( 16, 54)
( 17, 53)( 18, 52)( 19, 41)( 20, 40)( 21, 42)( 22, 38)( 23, 37)( 24, 39)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)
( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)
( 82,109)( 83,111)( 84,110)( 85,116)( 86,115)( 87,117)( 88,113)( 89,112)
( 90,114)( 91,131)( 92,130)( 93,132)( 94,128)( 95,127)( 96,129)( 97,135)
( 98,134)( 99,133)(100,122)(101,121)(102,123)(103,119)(104,118)(105,120)
(106,126)(107,125)(108,124)(137,138)(139,143)(140,142)(141,144)(145,158)
(146,157)(147,159)(148,155)(149,154)(150,156)(151,162)(152,161)(153,160)
(163,190)(164,192)(165,191)(166,197)(167,196)(168,198)(169,194)(170,193)
(171,195)(172,212)(173,211)(174,213)(175,209)(176,208)(177,210)(178,216)
(179,215)(180,214)(181,203)(182,202)(183,204)(184,200)(185,199)(186,201)
(187,207)(188,206)(189,205)(218,219)(220,224)(221,223)(222,225)(226,239)
(227,238)(228,240)(229,236)(230,235)(231,237)(232,243)(233,242)(234,241);;
s3 := (  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)(  8, 13)
(  9, 15)( 19, 23)( 20, 22)( 21, 24)( 25, 27)( 28, 64)( 29, 66)( 30, 65)
( 31, 71)( 32, 70)( 33, 72)( 34, 68)( 35, 67)( 36, 69)( 37, 55)( 38, 57)
( 39, 56)( 40, 62)( 41, 61)( 42, 63)( 43, 59)( 44, 58)( 45, 60)( 46, 77)
( 47, 76)( 48, 78)( 49, 74)( 50, 73)( 51, 75)( 52, 81)( 53, 80)( 54, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)( 89, 94)
( 90, 96)(100,104)(101,103)(102,105)(106,108)(109,145)(110,147)(111,146)
(112,152)(113,151)(114,153)(115,149)(116,148)(117,150)(118,136)(119,138)
(120,137)(121,143)(122,142)(123,144)(124,140)(125,139)(126,141)(127,158)
(128,157)(129,159)(130,155)(131,154)(132,156)(133,162)(134,161)(135,160)
(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)(170,175)
(171,177)(181,185)(182,184)(183,186)(187,189)(190,226)(191,228)(192,227)
(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,217)(200,219)
(201,218)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)(208,239)
(209,238)(210,240)(211,236)(212,235)(213,237)(214,243)(215,242)(216,241);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(243)!( 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)(109,190)(110,191)(111,192)(112,193)
(113,194)(114,195)(115,196)(116,197)(117,198)(118,199)(119,200)(120,201)
(121,202)(122,203)(123,204)(124,205)(125,206)(126,207)(127,208)(128,209)
(129,210)(130,211)(131,212)(132,213)(133,214)(134,215)(135,216)(136,217)
(137,218)(138,219)(139,220)(140,221)(141,222)(142,223)(143,224)(144,225)
(145,226)(146,227)(147,228)(148,229)(149,230)(150,231)(151,232)(152,233)
(153,234)(154,235)(155,236)(156,237)(157,238)(158,239)(159,240)(160,241)
(161,242)(162,243);
s1 := Sym(243)!(  1, 82)(  2, 83)(  3, 84)(  4, 85)(  5, 86)(  6, 87)(  7, 88)
(  8, 89)(  9, 90)( 10, 91)( 11, 92)( 12, 93)( 13, 94)( 14, 95)( 15, 96)
( 16, 97)( 17, 98)( 18, 99)( 19,100)( 20,101)( 21,102)( 22,103)( 23,104)
( 24,105)( 25,106)( 26,107)( 27,108)( 28,136)( 29,137)( 30,138)( 31,139)
( 32,140)( 33,141)( 34,142)( 35,143)( 36,144)( 37,145)( 38,146)( 39,147)
( 40,148)( 41,149)( 42,150)( 43,151)( 44,152)( 45,153)( 46,154)( 47,155)
( 48,156)( 49,157)( 50,158)( 51,159)( 52,160)( 53,161)( 54,162)( 55,109)
( 56,110)( 57,111)( 58,112)( 59,113)( 60,114)( 61,115)( 62,116)( 63,117)
( 64,118)( 65,119)( 66,120)( 67,121)( 68,122)( 69,123)( 70,124)( 71,125)
( 72,126)( 73,127)( 74,128)( 75,129)( 76,130)( 77,131)( 78,132)( 79,133)
( 80,134)( 81,135)(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);
s2 := Sym(243)!(  1, 28)(  2, 30)(  3, 29)(  4, 35)(  5, 34)(  6, 36)(  7, 32)
(  8, 31)(  9, 33)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)
( 16, 54)( 17, 53)( 18, 52)( 19, 41)( 20, 40)( 21, 42)( 22, 38)( 23, 37)
( 24, 39)( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 62)( 59, 61)( 60, 63)
( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)
( 72, 79)( 82,109)( 83,111)( 84,110)( 85,116)( 86,115)( 87,117)( 88,113)
( 89,112)( 90,114)( 91,131)( 92,130)( 93,132)( 94,128)( 95,127)( 96,129)
( 97,135)( 98,134)( 99,133)(100,122)(101,121)(102,123)(103,119)(104,118)
(105,120)(106,126)(107,125)(108,124)(137,138)(139,143)(140,142)(141,144)
(145,158)(146,157)(147,159)(148,155)(149,154)(150,156)(151,162)(152,161)
(153,160)(163,190)(164,192)(165,191)(166,197)(167,196)(168,198)(169,194)
(170,193)(171,195)(172,212)(173,211)(174,213)(175,209)(176,208)(177,210)
(178,216)(179,215)(180,214)(181,203)(182,202)(183,204)(184,200)(185,199)
(186,201)(187,207)(188,206)(189,205)(218,219)(220,224)(221,223)(222,225)
(226,239)(227,238)(228,240)(229,236)(230,235)(231,237)(232,243)(233,242)
(234,241);
s3 := Sym(243)!(  1, 10)(  2, 12)(  3, 11)(  4, 17)(  5, 16)(  6, 18)(  7, 14)
(  8, 13)(  9, 15)( 19, 23)( 20, 22)( 21, 24)( 25, 27)( 28, 64)( 29, 66)
( 30, 65)( 31, 71)( 32, 70)( 33, 72)( 34, 68)( 35, 67)( 36, 69)( 37, 55)
( 38, 57)( 39, 56)( 40, 62)( 41, 61)( 42, 63)( 43, 59)( 44, 58)( 45, 60)
( 46, 77)( 47, 76)( 48, 78)( 49, 74)( 50, 73)( 51, 75)( 52, 81)( 53, 80)
( 54, 79)( 82, 91)( 83, 93)( 84, 92)( 85, 98)( 86, 97)( 87, 99)( 88, 95)
( 89, 94)( 90, 96)(100,104)(101,103)(102,105)(106,108)(109,145)(110,147)
(111,146)(112,152)(113,151)(114,153)(115,149)(116,148)(117,150)(118,136)
(119,138)(120,137)(121,143)(122,142)(123,144)(124,140)(125,139)(126,141)
(127,158)(128,157)(129,159)(130,155)(131,154)(132,156)(133,162)(134,161)
(135,160)(163,172)(164,174)(165,173)(166,179)(167,178)(168,180)(169,176)
(170,175)(171,177)(181,185)(182,184)(183,186)(187,189)(190,226)(191,228)
(192,227)(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,217)
(200,219)(201,218)(202,224)(203,223)(204,225)(205,221)(206,220)(207,222)
(208,239)(209,238)(210,240)(211,236)(212,235)(213,237)(214,243)(215,242)
(216,241);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 
References : None.
to this polytope