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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}*1944b
if this polytope has a name.
Group : SmallGroup(1944,953)
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,  8)(  9, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 25)( 16, 24)
( 17, 26)( 18, 27)( 19, 29)( 20, 28)( 30, 57)( 31, 59)( 32, 58)( 33, 62)
( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 77)( 40, 76)( 41, 75)
( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)( 49, 67)
( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)( 84,185)
( 85,184)( 86,183)( 87,187)( 88,186)( 89,188)( 90,189)( 91,191)( 92,190)
( 93,176)( 94,175)( 95,174)( 96,178)( 97,177)( 98,179)( 99,180)(100,182)
(101,181)(102,167)(103,166)(104,165)(105,169)(106,168)(107,170)(108,171)
(109,173)(110,172)(111,239)(112,238)(113,237)(114,241)(115,240)(116,242)
(117,243)(118,245)(119,244)(120,230)(121,229)(122,228)(123,232)(124,231)
(125,233)(126,234)(127,236)(128,235)(129,221)(130,220)(131,219)(132,223)
(133,222)(134,224)(135,225)(136,227)(137,226)(138,212)(139,211)(140,210)
(141,214)(142,213)(143,215)(144,216)(145,218)(146,217)(147,203)(148,202)
(149,201)(150,205)(151,204)(152,206)(153,207)(154,209)(155,208)(156,194)
(157,193)(158,192)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199);;
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)( 57,221)( 58,220)
( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)( 66,238)
( 67,237)( 68,239)( 69,241)( 70,240)( 71,242)( 72,244)( 73,243)( 74,245)
( 75,229)( 76,228)( 77,230)( 78,232)( 79,231)( 80,233)( 81,235)( 82,234)
( 83,236)( 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)(138,140)(141,143)(144,146)(147,157)
(148,156)(149,158)(150,160)(151,159)(152,161)(153,163)(154,162)(155,164);;
s3 := (  6, 10)(  7, 11)(  8,  9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)( 25, 29)
( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)( 36, 62)
( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)( 44, 72)
( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)( 52, 83)
( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)( 96,100)
( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)(113,140)
(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)(121,148)
(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)(129,156)
(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)(137,160)
(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)(187,191)
(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)(198,224)
(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)(206,234)
(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)(214,245)
(215,243)(216,242)(217,240)(218,241);;
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*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, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!(  4,  5)(  6,  8)(  9, 10)( 12, 23)( 13, 22)( 14, 21)( 15, 25)
( 16, 24)( 17, 26)( 18, 27)( 19, 29)( 20, 28)( 30, 57)( 31, 59)( 32, 58)
( 33, 62)( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 77)( 40, 76)
( 41, 75)( 42, 79)( 43, 78)( 44, 80)( 45, 81)( 46, 83)( 47, 82)( 48, 68)
( 49, 67)( 50, 66)( 51, 70)( 52, 69)( 53, 71)( 54, 72)( 55, 74)( 56, 73)
( 84,185)( 85,184)( 86,183)( 87,187)( 88,186)( 89,188)( 90,189)( 91,191)
( 92,190)( 93,176)( 94,175)( 95,174)( 96,178)( 97,177)( 98,179)( 99,180)
(100,182)(101,181)(102,167)(103,166)(104,165)(105,169)(106,168)(107,170)
(108,171)(109,173)(110,172)(111,239)(112,238)(113,237)(114,241)(115,240)
(116,242)(117,243)(118,245)(119,244)(120,230)(121,229)(122,228)(123,232)
(124,231)(125,233)(126,234)(127,236)(128,235)(129,221)(130,220)(131,219)
(132,223)(133,222)(134,224)(135,225)(136,227)(137,226)(138,212)(139,211)
(140,210)(141,214)(142,213)(143,215)(144,216)(145,218)(146,217)(147,203)
(148,202)(149,201)(150,205)(151,204)(152,206)(153,207)(154,209)(155,208)
(156,194)(157,193)(158,192)(159,196)(160,195)(161,197)(162,198)(163,200)
(164,199);
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)( 57,221)
( 58,220)( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)
( 66,238)( 67,237)( 68,239)( 69,241)( 70,240)( 71,242)( 72,244)( 73,243)
( 74,245)( 75,229)( 76,228)( 77,230)( 78,232)( 79,231)( 80,233)( 81,235)
( 82,234)( 83,236)( 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)(138,140)(141,143)(144,146)
(147,157)(148,156)(149,158)(150,160)(151,159)(152,161)(153,163)(154,162)
(155,164);
s3 := Sym(245)!(  6, 10)(  7, 11)(  8,  9)( 15, 19)( 16, 20)( 17, 18)( 24, 28)
( 25, 29)( 26, 27)( 30, 57)( 31, 58)( 32, 59)( 33, 64)( 34, 65)( 35, 63)
( 36, 62)( 37, 60)( 38, 61)( 39, 66)( 40, 67)( 41, 68)( 42, 73)( 43, 74)
( 44, 72)( 45, 71)( 46, 69)( 47, 70)( 48, 75)( 49, 76)( 50, 77)( 51, 82)
( 52, 83)( 53, 81)( 54, 80)( 55, 78)( 56, 79)( 87, 91)( 88, 92)( 89, 90)
( 96,100)( 97,101)( 98, 99)(105,109)(106,110)(107,108)(111,138)(112,139)
(113,140)(114,145)(115,146)(116,144)(117,143)(118,141)(119,142)(120,147)
(121,148)(122,149)(123,154)(124,155)(125,153)(126,152)(127,150)(128,151)
(129,156)(130,157)(131,158)(132,163)(133,164)(134,162)(135,161)(136,159)
(137,160)(168,172)(169,173)(170,171)(177,181)(178,182)(179,180)(186,190)
(187,191)(188,189)(192,219)(193,220)(194,221)(195,226)(196,227)(197,225)
(198,224)(199,222)(200,223)(201,228)(202,229)(203,230)(204,235)(205,236)
(206,234)(207,233)(208,231)(209,232)(210,237)(211,238)(212,239)(213,244)
(214,245)(215,243)(216,242)(217,240)(218,241);
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, 
s3*s1*s2*s1*s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2 >; 
 

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