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Polytope of Type {2,27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,27,6}*1944c
if this polytope has a name.
Group : SmallGroup(1944,954)
Rank : 4
Schlafli Type : {2,27,6}
Number of vertices, edges, etc : 2, 81, 243, 18
Order of s0s1s2s3 : 54
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,9,6}*648a
   9-fold quotients : {2,9,6}*216, {2,3,6}*216
   27-fold quotients : {2,9,2}*72, {2,3,6}*72
   81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  4,  5)(  6,  7)(  9, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 24)( 16, 26)
( 17, 25)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)( 33, 61)
( 34, 60)( 35, 62)( 36, 65)( 37, 64)( 38, 63)( 39, 77)( 40, 76)( 41, 75)
( 42, 78)( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 83)( 48, 68)( 49, 67)
( 50, 66)( 51, 69)( 52, 71)( 53, 70)( 54, 73)( 55, 72)( 56, 74)( 84,185)
( 85,184)( 86,183)( 87,186)( 88,188)( 89,187)( 90,190)( 91,189)( 92,191)
( 93,176)( 94,175)( 95,174)( 96,177)( 97,179)( 98,178)( 99,181)(100,180)
(101,182)(102,167)(103,166)(104,165)(105,168)(106,170)(107,169)(108,172)
(109,171)(110,173)(111,239)(112,238)(113,237)(114,240)(115,242)(116,241)
(117,244)(118,243)(119,245)(120,230)(121,229)(122,228)(123,231)(124,233)
(125,232)(126,235)(127,234)(128,236)(129,221)(130,220)(131,219)(132,222)
(133,224)(134,223)(135,226)(136,225)(137,227)(138,212)(139,211)(140,210)
(141,213)(142,215)(143,214)(144,217)(145,216)(146,218)(147,203)(148,202)
(149,201)(150,204)(151,206)(152,205)(153,208)(154,207)(155,209)(156,194)
(157,193)(158,192)(159,195)(160,197)(161,196)(162,199)(163,198)(164,200);;
s2 := (  3,195)(  4,197)(  5,196)(  6,198)(  7,200)(  8,199)(  9,192)( 10,194)
( 11,193)( 12,215)( 13,214)( 14,213)( 15,218)( 16,217)( 17,216)( 18,212)
( 19,211)( 20,210)( 21,206)( 22,205)( 23,204)( 24,209)( 25,208)( 26,207)
( 27,203)( 28,202)( 29,201)( 30,171)( 31,173)( 32,172)( 33,165)( 34,167)
( 35,166)( 36,168)( 37,170)( 38,169)( 39,191)( 40,190)( 41,189)( 42,185)
( 43,184)( 44,183)( 45,188)( 46,187)( 47,186)( 48,182)( 49,181)( 50,180)
( 51,176)( 52,175)( 53,174)( 54,179)( 55,178)( 56,177)( 57,220)( 58,219)
( 59,221)( 60,223)( 61,222)( 62,224)( 63,226)( 64,225)( 65,227)( 66,237)
( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)( 72,243)( 73,245)( 74,244)
( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)( 80,232)( 81,234)( 82,236)
( 83,235)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)( 90,111)
( 91,113)( 92,112)( 93,134)( 94,133)( 95,132)( 96,137)( 97,136)( 98,135)
( 99,131)(100,130)(101,129)(102,125)(103,124)(104,123)(105,128)(106,127)
(107,126)(108,122)(109,121)(110,120)(138,139)(141,142)(144,145)(147,156)
(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)(155,163);;
s3 := (  6, 11)(  7,  9)(  8, 10)( 15, 20)( 16, 18)( 17, 19)( 24, 29)( 25, 27)
( 26, 28)( 30, 57)( 31, 58)( 32, 59)( 33, 65)( 34, 63)( 35, 64)( 36, 61)
( 37, 62)( 38, 60)( 39, 66)( 40, 67)( 41, 68)( 42, 74)( 43, 72)( 44, 73)
( 45, 70)( 46, 71)( 47, 69)( 48, 75)( 49, 76)( 50, 77)( 51, 83)( 52, 81)
( 53, 82)( 54, 79)( 55, 80)( 56, 78)( 87, 92)( 88, 90)( 89, 91)( 96,101)
( 97, 99)( 98,100)(105,110)(106,108)(107,109)(111,138)(112,139)(113,140)
(114,146)(115,144)(116,145)(117,142)(118,143)(119,141)(120,147)(121,148)
(122,149)(123,155)(124,153)(125,154)(126,151)(127,152)(128,150)(129,156)
(130,157)(131,158)(132,164)(133,162)(134,163)(135,160)(136,161)(137,159)
(168,173)(169,171)(170,172)(177,182)(178,180)(179,181)(186,191)(187,189)
(188,190)(192,219)(193,220)(194,221)(195,227)(196,225)(197,226)(198,223)
(199,224)(200,222)(201,228)(202,229)(203,230)(204,236)(205,234)(206,235)
(207,232)(208,233)(209,231)(210,237)(211,238)(212,239)(213,245)(214,243)
(215,244)(216,241)(217,242)(218,240);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s2*s1*s2*s1*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6,  7)(  9, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 24)
( 16, 26)( 17, 25)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)
( 33, 61)( 34, 60)( 35, 62)( 36, 65)( 37, 64)( 38, 63)( 39, 77)( 40, 76)
( 41, 75)( 42, 78)( 43, 80)( 44, 79)( 45, 82)( 46, 81)( 47, 83)( 48, 68)
( 49, 67)( 50, 66)( 51, 69)( 52, 71)( 53, 70)( 54, 73)( 55, 72)( 56, 74)
( 84,185)( 85,184)( 86,183)( 87,186)( 88,188)( 89,187)( 90,190)( 91,189)
( 92,191)( 93,176)( 94,175)( 95,174)( 96,177)( 97,179)( 98,178)( 99,181)
(100,180)(101,182)(102,167)(103,166)(104,165)(105,168)(106,170)(107,169)
(108,172)(109,171)(110,173)(111,239)(112,238)(113,237)(114,240)(115,242)
(116,241)(117,244)(118,243)(119,245)(120,230)(121,229)(122,228)(123,231)
(124,233)(125,232)(126,235)(127,234)(128,236)(129,221)(130,220)(131,219)
(132,222)(133,224)(134,223)(135,226)(136,225)(137,227)(138,212)(139,211)
(140,210)(141,213)(142,215)(143,214)(144,217)(145,216)(146,218)(147,203)
(148,202)(149,201)(150,204)(151,206)(152,205)(153,208)(154,207)(155,209)
(156,194)(157,193)(158,192)(159,195)(160,197)(161,196)(162,199)(163,198)
(164,200);
s2 := Sym(245)!(  3,195)(  4,197)(  5,196)(  6,198)(  7,200)(  8,199)(  9,192)
( 10,194)( 11,193)( 12,215)( 13,214)( 14,213)( 15,218)( 16,217)( 17,216)
( 18,212)( 19,211)( 20,210)( 21,206)( 22,205)( 23,204)( 24,209)( 25,208)
( 26,207)( 27,203)( 28,202)( 29,201)( 30,171)( 31,173)( 32,172)( 33,165)
( 34,167)( 35,166)( 36,168)( 37,170)( 38,169)( 39,191)( 40,190)( 41,189)
( 42,185)( 43,184)( 44,183)( 45,188)( 46,187)( 47,186)( 48,182)( 49,181)
( 50,180)( 51,176)( 52,175)( 53,174)( 54,179)( 55,178)( 56,177)( 57,220)
( 58,219)( 59,221)( 60,223)( 61,222)( 62,224)( 63,226)( 64,225)( 65,227)
( 66,237)( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)( 72,243)( 73,245)
( 74,244)( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)( 80,232)( 81,234)
( 82,236)( 83,235)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)
( 90,111)( 91,113)( 92,112)( 93,134)( 94,133)( 95,132)( 96,137)( 97,136)
( 98,135)( 99,131)(100,130)(101,129)(102,125)(103,124)(104,123)(105,128)
(106,127)(107,126)(108,122)(109,121)(110,120)(138,139)(141,142)(144,145)
(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162)(154,164)
(155,163);
s3 := Sym(245)!(  6, 11)(  7,  9)(  8, 10)( 15, 20)( 16, 18)( 17, 19)( 24, 29)
( 25, 27)( 26, 28)( 30, 57)( 31, 58)( 32, 59)( 33, 65)( 34, 63)( 35, 64)
( 36, 61)( 37, 62)( 38, 60)( 39, 66)( 40, 67)( 41, 68)( 42, 74)( 43, 72)
( 44, 73)( 45, 70)( 46, 71)( 47, 69)( 48, 75)( 49, 76)( 50, 77)( 51, 83)
( 52, 81)( 53, 82)( 54, 79)( 55, 80)( 56, 78)( 87, 92)( 88, 90)( 89, 91)
( 96,101)( 97, 99)( 98,100)(105,110)(106,108)(107,109)(111,138)(112,139)
(113,140)(114,146)(115,144)(116,145)(117,142)(118,143)(119,141)(120,147)
(121,148)(122,149)(123,155)(124,153)(125,154)(126,151)(127,152)(128,150)
(129,156)(130,157)(131,158)(132,164)(133,162)(134,163)(135,160)(136,161)
(137,159)(168,173)(169,171)(170,172)(177,182)(178,180)(179,181)(186,191)
(187,189)(188,190)(192,219)(193,220)(194,221)(195,227)(196,225)(197,226)
(198,223)(199,224)(200,222)(201,228)(202,229)(203,230)(204,236)(205,234)
(206,235)(207,232)(208,233)(209,231)(210,237)(211,238)(212,239)(213,245)
(214,243)(215,244)(216,241)(217,242)(218,240);
poly := sub<Sym(245)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s2*s1*s2*s1*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2 >; 
 

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