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Polytope of Type {6,82}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,82}*984
Also Known As : {6,82|2}. if this polytope has another name.
Group : SmallGroup(984,41)
Rank : 3
Schlafli Type : {6,82}
Number of vertices, edges, etc : 6, 246, 82
Order of s0s1s2 : 246
Order of s0s1s2s1 : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,82,2} of size 1968
Vertex Figure Of :
   {2,6,82} of size 1968
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,82}*328
   6-fold quotients : {2,41}*164
   41-fold quotients : {6,2}*24
   82-fold quotients : {3,2}*12
   123-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,82}*1968, {6,164}*1968a
Permutation Representation (GAP) :
s0 := ( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)( 49, 90)
( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)( 57, 98)
( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)( 65,106)
( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)( 73,114)
( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)( 81,122)
( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)(171,212)
(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)(179,220)
(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)(187,228)
(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)(195,236)
(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)(203,244)
(204,245)(205,246);;
s1 := (  1, 42)(  2, 82)(  3, 81)(  4, 80)(  5, 79)(  6, 78)(  7, 77)(  8, 76)
(  9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)( 16, 68)
( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)( 24, 60)
( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)( 32, 52)
( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)( 40, 44)
( 41, 43)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)( 90,117)
( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)
( 99,108)(100,107)(101,106)(102,105)(103,104)(124,165)(125,205)(126,204)
(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)(134,196)
(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)(142,188)
(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)(150,180)
(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)(158,172)
(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(207,246)(208,245)
(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)
(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)(224,229)
(225,228)(226,227);;
s2 := (  1,125)(  2,124)(  3,164)(  4,163)(  5,162)(  6,161)(  7,160)(  8,159)
(  9,158)( 10,157)( 11,156)( 12,155)( 13,154)( 14,153)( 15,152)( 16,151)
( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,144)( 24,143)
( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)( 32,135)
( 33,134)( 34,133)( 35,132)( 36,131)( 37,130)( 38,129)( 39,128)( 40,127)
( 41,126)( 42,166)( 43,165)( 44,205)( 45,204)( 46,203)( 47,202)( 48,201)
( 49,200)( 50,199)( 51,198)( 52,197)( 53,196)( 54,195)( 55,194)( 56,193)
( 57,192)( 58,191)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)( 64,185)
( 65,184)( 66,183)( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)( 72,177)
( 73,176)( 74,175)( 75,174)( 76,173)( 77,172)( 78,171)( 79,170)( 80,169)
( 81,168)( 82,167)( 83,207)( 84,206)( 85,246)( 86,245)( 87,244)( 88,243)
( 89,242)( 90,241)( 91,240)( 92,239)( 93,238)( 94,237)( 95,236)( 96,235)
( 97,234)( 98,233)( 99,232)(100,231)(101,230)(102,229)(103,228)(104,227)
(105,226)(106,225)(107,224)(108,223)(109,222)(110,221)(111,220)(112,219)
(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)(119,212)(120,211)
(121,210)(122,209)(123,208);;
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*s2*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*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(246)!( 42, 83)( 43, 84)( 44, 85)( 45, 86)( 46, 87)( 47, 88)( 48, 89)
( 49, 90)( 50, 91)( 51, 92)( 52, 93)( 53, 94)( 54, 95)( 55, 96)( 56, 97)
( 57, 98)( 58, 99)( 59,100)( 60,101)( 61,102)( 62,103)( 63,104)( 64,105)
( 65,106)( 66,107)( 67,108)( 68,109)( 69,110)( 70,111)( 71,112)( 72,113)
( 73,114)( 74,115)( 75,116)( 76,117)( 77,118)( 78,119)( 79,120)( 80,121)
( 81,122)( 82,123)(165,206)(166,207)(167,208)(168,209)(169,210)(170,211)
(171,212)(172,213)(173,214)(174,215)(175,216)(176,217)(177,218)(178,219)
(179,220)(180,221)(181,222)(182,223)(183,224)(184,225)(185,226)(186,227)
(187,228)(188,229)(189,230)(190,231)(191,232)(192,233)(193,234)(194,235)
(195,236)(196,237)(197,238)(198,239)(199,240)(200,241)(201,242)(202,243)
(203,244)(204,245)(205,246);
s1 := Sym(246)!(  1, 42)(  2, 82)(  3, 81)(  4, 80)(  5, 79)(  6, 78)(  7, 77)
(  8, 76)(  9, 75)( 10, 74)( 11, 73)( 12, 72)( 13, 71)( 14, 70)( 15, 69)
( 16, 68)( 17, 67)( 18, 66)( 19, 65)( 20, 64)( 21, 63)( 22, 62)( 23, 61)
( 24, 60)( 25, 59)( 26, 58)( 27, 57)( 28, 56)( 29, 55)( 30, 54)( 31, 53)
( 32, 52)( 33, 51)( 34, 50)( 35, 49)( 36, 48)( 37, 47)( 38, 46)( 39, 45)
( 40, 44)( 41, 43)( 84,123)( 85,122)( 86,121)( 87,120)( 88,119)( 89,118)
( 90,117)( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)
( 98,109)( 99,108)(100,107)(101,106)(102,105)(103,104)(124,165)(125,205)
(126,204)(127,203)(128,202)(129,201)(130,200)(131,199)(132,198)(133,197)
(134,196)(135,195)(136,194)(137,193)(138,192)(139,191)(140,190)(141,189)
(142,188)(143,187)(144,186)(145,185)(146,184)(147,183)(148,182)(149,181)
(150,180)(151,179)(152,178)(153,177)(154,176)(155,175)(156,174)(157,173)
(158,172)(159,171)(160,170)(161,169)(162,168)(163,167)(164,166)(207,246)
(208,245)(209,244)(210,243)(211,242)(212,241)(213,240)(214,239)(215,238)
(216,237)(217,236)(218,235)(219,234)(220,233)(221,232)(222,231)(223,230)
(224,229)(225,228)(226,227);
s2 := Sym(246)!(  1,125)(  2,124)(  3,164)(  4,163)(  5,162)(  6,161)(  7,160)
(  8,159)(  9,158)( 10,157)( 11,156)( 12,155)( 13,154)( 14,153)( 15,152)
( 16,151)( 17,150)( 18,149)( 19,148)( 20,147)( 21,146)( 22,145)( 23,144)
( 24,143)( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)
( 32,135)( 33,134)( 34,133)( 35,132)( 36,131)( 37,130)( 38,129)( 39,128)
( 40,127)( 41,126)( 42,166)( 43,165)( 44,205)( 45,204)( 46,203)( 47,202)
( 48,201)( 49,200)( 50,199)( 51,198)( 52,197)( 53,196)( 54,195)( 55,194)
( 56,193)( 57,192)( 58,191)( 59,190)( 60,189)( 61,188)( 62,187)( 63,186)
( 64,185)( 65,184)( 66,183)( 67,182)( 68,181)( 69,180)( 70,179)( 71,178)
( 72,177)( 73,176)( 74,175)( 75,174)( 76,173)( 77,172)( 78,171)( 79,170)
( 80,169)( 81,168)( 82,167)( 83,207)( 84,206)( 85,246)( 86,245)( 87,244)
( 88,243)( 89,242)( 90,241)( 91,240)( 92,239)( 93,238)( 94,237)( 95,236)
( 96,235)( 97,234)( 98,233)( 99,232)(100,231)(101,230)(102,229)(103,228)
(104,227)(105,226)(106,225)(107,224)(108,223)(109,222)(110,221)(111,220)
(112,219)(113,218)(114,217)(115,216)(116,215)(117,214)(118,213)(119,212)
(120,211)(121,210)(122,209)(123,208);
poly := sub<Sym(246)|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*s2*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*s1*s2 >; 
 
References : None.
to this polytope