Polytope of Type {27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,6}*972a
if this polytope has a name.
Group : SmallGroup(972,107)
Rank : 3
Schlafli Type : {27,6}
Number of vertices, edges, etc : 81, 243, 18
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {27,6,2} of size 1944
Vertex Figure Of :
   {2,27,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,6}*324a, {27,6}*324
   9-fold quotients : {27,2}*108, {9,6}*108, {3,6}*108
   27-fold quotients : {9,2}*36, {3,6}*36
   81-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {54,6}*1944a
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope