Polytope of Type {6,27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,27,6}*1944
if this polytope has a name.
Group : SmallGroup(1944,2343)
Rank : 4
Schlafli Type : {6,27,6}
Number of vertices, edges, etc : 6, 81, 81, 6
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
   Self-Dual
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 : {2,27,6}*648, {6,27,2}*648, {6,9,6}*648
   9-fold quotients : {2,27,2}*216, {2,9,6}*216, {6,9,2}*216, {6,3,6}*216
   27-fold quotients : {2,9,2}*72, {2,3,6}*72, {6,3,2}*72
   81-fold quotients : {2,3,2}*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, 84)(  3, 83)(  4, 89)(  5, 88)(  6, 90)(  7, 86)(  8, 85)
(  9, 87)( 10,104)( 11,103)( 12,105)( 13,101)( 14,100)( 15,102)( 16,108)
( 17,107)( 18,106)( 19, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)( 24, 93)
( 25, 99)( 26, 98)( 27, 97)( 28,136)( 29,138)( 30,137)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,158)( 38,157)( 39,159)( 40,155)
( 41,154)( 42,156)( 43,162)( 44,161)( 45,160)( 46,149)( 47,148)( 48,150)
( 49,146)( 50,145)( 51,147)( 52,153)( 53,152)( 54,151)( 55,109)( 56,111)
( 57,110)( 58,116)( 59,115)( 60,117)( 61,113)( 62,112)( 63,114)( 64,131)
( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,135)( 71,134)( 72,133)
( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,126)( 80,125)
( 81,124)(164,165)(166,170)(167,169)(168,171)(172,185)(173,184)(174,186)
(175,182)(176,181)(177,183)(178,189)(179,188)(180,187)(190,217)(191,219)
(192,218)(193,224)(194,223)(195,225)(196,221)(197,220)(198,222)(199,239)
(200,238)(201,240)(202,236)(203,235)(204,237)(205,243)(206,242)(207,241)
(208,230)(209,229)(210,231)(211,227)(212,226)(213,228)(214,234)(215,233)
(216,232);;
s2 := (  1, 37)(  2, 39)(  3, 38)(  4, 44)(  5, 43)(  6, 45)(  7, 41)(  8, 40)
(  9, 42)( 10, 28)( 11, 30)( 12, 29)( 13, 35)( 14, 34)( 15, 36)( 16, 32)
( 17, 31)( 18, 33)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)( 24, 48)
( 25, 54)( 26, 53)( 27, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 71)( 59, 70)
( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 77)( 74, 76)( 75, 78)( 79, 81)
( 82,199)( 83,201)( 84,200)( 85,206)( 86,205)( 87,207)( 88,203)( 89,202)
( 90,204)( 91,190)( 92,192)( 93,191)( 94,197)( 95,196)( 96,198)( 97,194)
( 98,193)( 99,195)(100,212)(101,211)(102,213)(103,209)(104,208)(105,210)
(106,216)(107,215)(108,214)(109,172)(110,174)(111,173)(112,179)(113,178)
(114,180)(115,176)(116,175)(117,177)(118,163)(119,165)(120,164)(121,170)
(122,169)(123,171)(124,167)(125,166)(126,168)(127,185)(128,184)(129,186)
(130,182)(131,181)(132,183)(133,189)(134,188)(135,187)(136,226)(137,228)
(138,227)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,217)
(146,219)(147,218)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)
(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,243)(161,242)
(162,241);;
s3 := ( 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)(109,136)(110,137)(111,138)(112,139)(113,140)
(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)
(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)
(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(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);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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, 84)(  3, 83)(  4, 89)(  5, 88)(  6, 90)(  7, 86)
(  8, 85)(  9, 87)( 10,104)( 11,103)( 12,105)( 13,101)( 14,100)( 15,102)
( 16,108)( 17,107)( 18,106)( 19, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)
( 24, 93)( 25, 99)( 26, 98)( 27, 97)( 28,136)( 29,138)( 30,137)( 31,143)
( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,158)( 38,157)( 39,159)
( 40,155)( 41,154)( 42,156)( 43,162)( 44,161)( 45,160)( 46,149)( 47,148)
( 48,150)( 49,146)( 50,145)( 51,147)( 52,153)( 53,152)( 54,151)( 55,109)
( 56,111)( 57,110)( 58,116)( 59,115)( 60,117)( 61,113)( 62,112)( 63,114)
( 64,131)( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,135)( 71,134)
( 72,133)( 73,122)( 74,121)( 75,123)( 76,119)( 77,118)( 78,120)( 79,126)
( 80,125)( 81,124)(164,165)(166,170)(167,169)(168,171)(172,185)(173,184)
(174,186)(175,182)(176,181)(177,183)(178,189)(179,188)(180,187)(190,217)
(191,219)(192,218)(193,224)(194,223)(195,225)(196,221)(197,220)(198,222)
(199,239)(200,238)(201,240)(202,236)(203,235)(204,237)(205,243)(206,242)
(207,241)(208,230)(209,229)(210,231)(211,227)(212,226)(213,228)(214,234)
(215,233)(216,232);
s2 := Sym(243)!(  1, 37)(  2, 39)(  3, 38)(  4, 44)(  5, 43)(  6, 45)(  7, 41)
(  8, 40)(  9, 42)( 10, 28)( 11, 30)( 12, 29)( 13, 35)( 14, 34)( 15, 36)
( 16, 32)( 17, 31)( 18, 33)( 19, 50)( 20, 49)( 21, 51)( 22, 47)( 23, 46)
( 24, 48)( 25, 54)( 26, 53)( 27, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 71)
( 59, 70)( 60, 72)( 61, 68)( 62, 67)( 63, 69)( 73, 77)( 74, 76)( 75, 78)
( 79, 81)( 82,199)( 83,201)( 84,200)( 85,206)( 86,205)( 87,207)( 88,203)
( 89,202)( 90,204)( 91,190)( 92,192)( 93,191)( 94,197)( 95,196)( 96,198)
( 97,194)( 98,193)( 99,195)(100,212)(101,211)(102,213)(103,209)(104,208)
(105,210)(106,216)(107,215)(108,214)(109,172)(110,174)(111,173)(112,179)
(113,178)(114,180)(115,176)(116,175)(117,177)(118,163)(119,165)(120,164)
(121,170)(122,169)(123,171)(124,167)(125,166)(126,168)(127,185)(128,184)
(129,186)(130,182)(131,181)(132,183)(133,189)(134,188)(135,187)(136,226)
(137,228)(138,227)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)
(145,217)(146,219)(147,218)(148,224)(149,223)(150,225)(151,221)(152,220)
(153,222)(154,239)(155,238)(156,240)(157,236)(158,235)(159,237)(160,243)
(161,242)(162,241);
s3 := 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)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)
(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(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);
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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope