Polytope of Type {27,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,6}*972c
if this polytope has a name.
Group : SmallGroup(972,113)
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
   9-fold quotients : {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}*1944e
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope