Polytope of Type {27,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,6,2}*1944b
if this polytope has a name.
Group : SmallGroup(1944,953)
Rank : 4
Schlafli Type : {27,6,2}
Number of vertices, edges, etc : 81, 243, 18, 2
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 : {9,6,2}*648a
   9-fold quotients : {9,6,2}*216, {3,6,2}*216
   27-fold quotients : {9,2,2}*72, {3,6,2}*72
   81-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2,  3)(  4,  6)(  7,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 23)( 14, 22)
( 15, 24)( 16, 25)( 17, 27)( 18, 26)( 28, 55)( 29, 57)( 30, 56)( 31, 60)
( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 75)( 38, 74)( 39, 73)
( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)( 47, 65)
( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)( 82,183)
( 83,182)( 84,181)( 85,185)( 86,184)( 87,186)( 88,187)( 89,189)( 90,188)
( 91,174)( 92,173)( 93,172)( 94,176)( 95,175)( 96,177)( 97,178)( 98,180)
( 99,179)(100,165)(101,164)(102,163)(103,167)(104,166)(105,168)(106,169)
(107,171)(108,170)(109,237)(110,236)(111,235)(112,239)(113,238)(114,240)
(115,241)(116,243)(117,242)(118,228)(119,227)(120,226)(121,230)(122,229)
(123,231)(124,232)(125,234)(126,233)(127,219)(128,218)(129,217)(130,221)
(131,220)(132,222)(133,223)(134,225)(135,224)(136,210)(137,209)(138,208)
(139,212)(140,211)(141,213)(142,214)(143,216)(144,215)(145,201)(146,200)
(147,199)(148,203)(149,202)(150,204)(151,205)(152,207)(153,206)(154,192)
(155,191)(156,190)(157,194)(158,193)(159,195)(160,196)(161,198)(162,197);;
s1 := (  1,193)(  2,195)(  3,194)(  4,196)(  5,198)(  6,197)(  7,190)(  8,192)
(  9,191)( 10,213)( 11,212)( 12,211)( 13,216)( 14,215)( 15,214)( 16,210)
( 17,209)( 18,208)( 19,204)( 20,203)( 21,202)( 22,207)( 23,206)( 24,205)
( 25,201)( 26,200)( 27,199)( 28,169)( 29,171)( 30,170)( 31,163)( 32,165)
( 33,164)( 34,166)( 35,168)( 36,167)( 37,189)( 38,188)( 39,187)( 40,183)
( 41,182)( 42,181)( 43,186)( 44,185)( 45,184)( 46,180)( 47,179)( 48,178)
( 49,174)( 50,173)( 51,172)( 52,177)( 53,176)( 54,175)( 55,219)( 56,218)
( 57,217)( 58,222)( 59,221)( 60,220)( 61,225)( 62,224)( 63,223)( 64,236)
( 65,235)( 66,237)( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)( 72,243)
( 73,227)( 74,226)( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)( 80,232)
( 81,234)( 82,112)( 83,114)( 84,113)( 85,115)( 86,117)( 87,116)( 88,109)
( 89,111)( 90,110)( 91,132)( 92,131)( 93,130)( 94,135)( 95,134)( 96,133)
( 97,129)( 98,128)( 99,127)(100,123)(101,122)(102,121)(103,126)(104,125)
(105,124)(106,120)(107,119)(108,118)(136,138)(139,141)(142,144)(145,155)
(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)(153,162);;
s2 := (  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)( 23, 27)
( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)( 34, 60)
( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)( 42, 70)
( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)( 50, 81)
( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)( 94, 98)
( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)(111,138)
(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)(119,146)
(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)(127,154)
(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)(135,158)
(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)(185,189)
(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)(196,222)
(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)(204,232)
(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)(212,243)
(213,241)(214,240)(215,238)(216,239);;
s3 := (244,245);;
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*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(245)!(  2,  3)(  4,  6)(  7,  8)( 10, 21)( 11, 20)( 12, 19)( 13, 23)
( 14, 22)( 15, 24)( 16, 25)( 17, 27)( 18, 26)( 28, 55)( 29, 57)( 30, 56)
( 31, 60)( 32, 59)( 33, 58)( 34, 62)( 35, 61)( 36, 63)( 37, 75)( 38, 74)
( 39, 73)( 40, 77)( 41, 76)( 42, 78)( 43, 79)( 44, 81)( 45, 80)( 46, 66)
( 47, 65)( 48, 64)( 49, 68)( 50, 67)( 51, 69)( 52, 70)( 53, 72)( 54, 71)
( 82,183)( 83,182)( 84,181)( 85,185)( 86,184)( 87,186)( 88,187)( 89,189)
( 90,188)( 91,174)( 92,173)( 93,172)( 94,176)( 95,175)( 96,177)( 97,178)
( 98,180)( 99,179)(100,165)(101,164)(102,163)(103,167)(104,166)(105,168)
(106,169)(107,171)(108,170)(109,237)(110,236)(111,235)(112,239)(113,238)
(114,240)(115,241)(116,243)(117,242)(118,228)(119,227)(120,226)(121,230)
(122,229)(123,231)(124,232)(125,234)(126,233)(127,219)(128,218)(129,217)
(130,221)(131,220)(132,222)(133,223)(134,225)(135,224)(136,210)(137,209)
(138,208)(139,212)(140,211)(141,213)(142,214)(143,216)(144,215)(145,201)
(146,200)(147,199)(148,203)(149,202)(150,204)(151,205)(152,207)(153,206)
(154,192)(155,191)(156,190)(157,194)(158,193)(159,195)(160,196)(161,198)
(162,197);
s1 := Sym(245)!(  1,193)(  2,195)(  3,194)(  4,196)(  5,198)(  6,197)(  7,190)
(  8,192)(  9,191)( 10,213)( 11,212)( 12,211)( 13,216)( 14,215)( 15,214)
( 16,210)( 17,209)( 18,208)( 19,204)( 20,203)( 21,202)( 22,207)( 23,206)
( 24,205)( 25,201)( 26,200)( 27,199)( 28,169)( 29,171)( 30,170)( 31,163)
( 32,165)( 33,164)( 34,166)( 35,168)( 36,167)( 37,189)( 38,188)( 39,187)
( 40,183)( 41,182)( 42,181)( 43,186)( 44,185)( 45,184)( 46,180)( 47,179)
( 48,178)( 49,174)( 50,173)( 51,172)( 52,177)( 53,176)( 54,175)( 55,219)
( 56,218)( 57,217)( 58,222)( 59,221)( 60,220)( 61,225)( 62,224)( 63,223)
( 64,236)( 65,235)( 66,237)( 67,239)( 68,238)( 69,240)( 70,242)( 71,241)
( 72,243)( 73,227)( 74,226)( 75,228)( 76,230)( 77,229)( 78,231)( 79,233)
( 80,232)( 81,234)( 82,112)( 83,114)( 84,113)( 85,115)( 86,117)( 87,116)
( 88,109)( 89,111)( 90,110)( 91,132)( 92,131)( 93,130)( 94,135)( 95,134)
( 96,133)( 97,129)( 98,128)( 99,127)(100,123)(101,122)(102,121)(103,126)
(104,125)(105,124)(106,120)(107,119)(108,118)(136,138)(139,141)(142,144)
(145,155)(146,154)(147,156)(148,158)(149,157)(150,159)(151,161)(152,160)
(153,162);
s2 := Sym(245)!(  4,  8)(  5,  9)(  6,  7)( 13, 17)( 14, 18)( 15, 16)( 22, 26)
( 23, 27)( 24, 25)( 28, 55)( 29, 56)( 30, 57)( 31, 62)( 32, 63)( 33, 61)
( 34, 60)( 35, 58)( 36, 59)( 37, 64)( 38, 65)( 39, 66)( 40, 71)( 41, 72)
( 42, 70)( 43, 69)( 44, 67)( 45, 68)( 46, 73)( 47, 74)( 48, 75)( 49, 80)
( 50, 81)( 51, 79)( 52, 78)( 53, 76)( 54, 77)( 85, 89)( 86, 90)( 87, 88)
( 94, 98)( 95, 99)( 96, 97)(103,107)(104,108)(105,106)(109,136)(110,137)
(111,138)(112,143)(113,144)(114,142)(115,141)(116,139)(117,140)(118,145)
(119,146)(120,147)(121,152)(122,153)(123,151)(124,150)(125,148)(126,149)
(127,154)(128,155)(129,156)(130,161)(131,162)(132,160)(133,159)(134,157)
(135,158)(166,170)(167,171)(168,169)(175,179)(176,180)(177,178)(184,188)
(185,189)(186,187)(190,217)(191,218)(192,219)(193,224)(194,225)(195,223)
(196,222)(197,220)(198,221)(199,226)(200,227)(201,228)(202,233)(203,234)
(204,232)(205,231)(206,229)(207,230)(208,235)(209,236)(210,237)(211,242)
(212,243)(213,241)(214,240)(215,238)(216,239);
s3 := Sym(245)!(244,245);
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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

to this polytope