Polytope of Type {6,81}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,81}*972
if this polytope has a name.
Group : SmallGroup(972,114)
Rank : 3
Schlafli Type : {6,81}
Number of vertices, edges, etc : 6, 243, 81
Order of s₀s₁s₂ : 162
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,81,2} of size 1944
Vertex Figure Of :
   {2,6,81} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,81}*324, {6,27}*324
   9-fold quotients : {2,27}*108, {6,9}*108
   27-fold quotients : {2,9}*36, {6,3}*36
   81-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,162}*1944b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope