Polytope of Type {82,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {82,6}*984
Also Known As : {82,6|2}. if this polytope has another name.
Group : SmallGroup(984,41)
Rank : 3
Schlafli Type : {82,6}
Number of vertices, edges, etc : 82, 246, 6
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 :
   {82,6,2} of size 1968
Vertex Figure Of :
   {2,82,6} of size 1968
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {82,2}*328
   6-fold quotients : {41,2}*164
   41-fold quotients : {2,6}*24
   82-fold quotients : {2,3}*12
   123-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {82,12}*1968, {164,6}*1968a
Permutation Representation (GAP) :

s0 := (  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)(  9, 34)
( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)( 17, 26)
( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)( 46, 79)
( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)( 54, 71)
( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)( 62, 63)
( 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)(125,164)(126,163)(127,162)(128,161)
(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)(136,153)
(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)(144,145)
(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)(173,198)
(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)(181,190)
(182,189)(183,188)(184,187)(185,186)(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);;
s1 := (  1,  2)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)(  8, 36)(  9, 35)
( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 29)( 16, 28)( 17, 27)
( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 42, 84)( 43, 83)( 44,123)( 45,122)
( 46,121)( 47,120)( 48,119)( 49,118)( 50,117)( 51,116)( 52,115)( 53,114)
( 54,113)( 55,112)( 56,111)( 57,110)( 58,109)( 59,108)( 60,107)( 61,106)
( 62,105)( 63,104)( 64,103)( 65,102)( 66,101)( 67,100)( 68, 99)( 69, 98)
( 70, 97)( 71, 96)( 72, 95)( 73, 94)( 74, 93)( 75, 92)( 76, 91)( 77, 90)
( 78, 89)( 79, 88)( 80, 87)( 81, 86)( 82, 85)(124,125)(126,164)(127,163)
(128,162)(129,161)(130,160)(131,159)(132,158)(133,157)(134,156)(135,155)
(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)(143,147)
(144,146)(165,207)(166,206)(167,246)(168,245)(169,244)(170,243)(171,242)
(172,241)(173,240)(174,239)(175,238)(176,237)(177,236)(178,235)(179,234)
(180,233)(181,232)(182,231)(183,230)(184,229)(185,228)(186,227)(187,226)
(188,225)(189,224)(190,223)(191,222)(192,221)(193,220)(194,219)(195,218)
(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)(203,210)
(204,209)(205,208);;
s2 := (  1,165)(  2,166)(  3,167)(  4,168)(  5,169)(  6,170)(  7,171)(  8,172)
(  9,173)( 10,174)( 11,175)( 12,176)( 13,177)( 14,178)( 15,179)( 16,180)
( 17,181)( 18,182)( 19,183)( 20,184)( 21,185)( 22,186)( 23,187)( 24,188)
( 25,189)( 26,190)( 27,191)( 28,192)( 29,193)( 30,194)( 31,195)( 32,196)
( 33,197)( 34,198)( 35,199)( 36,200)( 37,201)( 38,202)( 39,203)( 40,204)
( 41,205)( 42,124)( 43,125)( 44,126)( 45,127)( 46,128)( 47,129)( 48,130)
( 49,131)( 50,132)( 51,133)( 52,134)( 53,135)( 54,136)( 55,137)( 56,138)
( 57,139)( 58,140)( 59,141)( 60,142)( 61,143)( 62,144)( 63,145)( 64,146)
( 65,147)( 66,148)( 67,149)( 68,150)( 69,151)( 70,152)( 71,153)( 72,154)
( 73,155)( 74,156)( 75,157)( 76,158)( 77,159)( 78,160)( 79,161)( 80,162)
( 81,163)( 82,164)( 83,206)( 84,207)( 85,208)( 86,209)( 87,210)( 88,211)
( 89,212)( 90,213)( 91,214)( 92,215)( 93,216)( 94,217)( 95,218)( 96,219)
( 97,220)( 98,221)( 99,222)(100,223)(101,224)(102,225)(103,226)(104,227)
(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)(112,235)
(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)(120,243)
(121,244)(122,245)(123,246);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(246)!(  2, 41)(  3, 40)(  4, 39)(  5, 38)(  6, 37)(  7, 36)(  8, 35)
(  9, 34)( 10, 33)( 11, 32)( 12, 31)( 13, 30)( 14, 29)( 15, 28)( 16, 27)
( 17, 26)( 18, 25)( 19, 24)( 20, 23)( 21, 22)( 43, 82)( 44, 81)( 45, 80)
( 46, 79)( 47, 78)( 48, 77)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 72)
( 54, 71)( 55, 70)( 56, 69)( 57, 68)( 58, 67)( 59, 66)( 60, 65)( 61, 64)
( 62, 63)( 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)(125,164)(126,163)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154)
(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)(143,146)
(144,145)(166,205)(167,204)(168,203)(169,202)(170,201)(171,200)(172,199)
(173,198)(174,197)(175,196)(176,195)(177,194)(178,193)(179,192)(180,191)
(181,190)(182,189)(183,188)(184,187)(185,186)(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);
s1 := Sym(246)!(  1,  2)(  3, 41)(  4, 40)(  5, 39)(  6, 38)(  7, 37)(  8, 36)
(  9, 35)( 10, 34)( 11, 33)( 12, 32)( 13, 31)( 14, 30)( 15, 29)( 16, 28)
( 17, 27)( 18, 26)( 19, 25)( 20, 24)( 21, 23)( 42, 84)( 43, 83)( 44,123)
( 45,122)( 46,121)( 47,120)( 48,119)( 49,118)( 50,117)( 51,116)( 52,115)
( 53,114)( 54,113)( 55,112)( 56,111)( 57,110)( 58,109)( 59,108)( 60,107)
( 61,106)( 62,105)( 63,104)( 64,103)( 65,102)( 66,101)( 67,100)( 68, 99)
( 69, 98)( 70, 97)( 71, 96)( 72, 95)( 73, 94)( 74, 93)( 75, 92)( 76, 91)
( 77, 90)( 78, 89)( 79, 88)( 80, 87)( 81, 86)( 82, 85)(124,125)(126,164)
(127,163)(128,162)(129,161)(130,160)(131,159)(132,158)(133,157)(134,156)
(135,155)(136,154)(137,153)(138,152)(139,151)(140,150)(141,149)(142,148)
(143,147)(144,146)(165,207)(166,206)(167,246)(168,245)(169,244)(170,243)
(171,242)(172,241)(173,240)(174,239)(175,238)(176,237)(177,236)(178,235)
(179,234)(180,233)(181,232)(182,231)(183,230)(184,229)(185,228)(186,227)
(187,226)(188,225)(189,224)(190,223)(191,222)(192,221)(193,220)(194,219)
(195,218)(196,217)(197,216)(198,215)(199,214)(200,213)(201,212)(202,211)
(203,210)(204,209)(205,208);
s2 := Sym(246)!(  1,165)(  2,166)(  3,167)(  4,168)(  5,169)(  6,170)(  7,171)
(  8,172)(  9,173)( 10,174)( 11,175)( 12,176)( 13,177)( 14,178)( 15,179)
( 16,180)( 17,181)( 18,182)( 19,183)( 20,184)( 21,185)( 22,186)( 23,187)
( 24,188)( 25,189)( 26,190)( 27,191)( 28,192)( 29,193)( 30,194)( 31,195)
( 32,196)( 33,197)( 34,198)( 35,199)( 36,200)( 37,201)( 38,202)( 39,203)
( 40,204)( 41,205)( 42,124)( 43,125)( 44,126)( 45,127)( 46,128)( 47,129)
( 48,130)( 49,131)( 50,132)( 51,133)( 52,134)( 53,135)( 54,136)( 55,137)
( 56,138)( 57,139)( 58,140)( 59,141)( 60,142)( 61,143)( 62,144)( 63,145)
( 64,146)( 65,147)( 66,148)( 67,149)( 68,150)( 69,151)( 70,152)( 71,153)
( 72,154)( 73,155)( 74,156)( 75,157)( 76,158)( 77,159)( 78,160)( 79,161)
( 80,162)( 81,163)( 82,164)( 83,206)( 84,207)( 85,208)( 86,209)( 87,210)
( 88,211)( 89,212)( 90,213)( 91,214)( 92,215)( 93,216)( 94,217)( 95,218)
( 96,219)( 97,220)( 98,221)( 99,222)(100,223)(101,224)(102,225)(103,226)
(104,227)(105,228)(106,229)(107,230)(108,231)(109,232)(110,233)(111,234)
(112,235)(113,236)(114,237)(115,238)(116,239)(117,240)(118,241)(119,242)
(120,243)(121,244)(122,245)(123,246);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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*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*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*s0*s1 >;

References : None.
to this polytope