Polytope of Type {2,82,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,82,6}*1968
if this polytope has a name.
Group : SmallGroup(1968,197)
Rank : 4
Schlafli Type : {2,82,6}
Number of vertices, edges, etc : 2, 82, 246, 6
Order of s₀s₁s₂s₃ : 246
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,82,2}*656
   6-fold quotients : {2,41,2}*328
   41-fold quotients : {2,2,6}*48
   82-fold quotients : {2,2,3}*24
   123-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (  4, 43)(  5, 42)(  6, 41)(  7, 40)(  8, 39)(  9, 38)( 10, 37)( 11, 36)
( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16, 31)( 17, 30)( 18, 29)( 19, 28)
( 20, 27)( 21, 26)( 22, 25)( 23, 24)( 45, 84)( 46, 83)( 47, 82)( 48, 81)
( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)( 56, 73)
( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 61, 68)( 62, 67)( 63, 66)( 64, 65)
( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,120)( 92,119)( 93,118)
( 94,117)( 95,116)( 96,115)( 97,114)( 98,113)( 99,112)(100,111)(101,110)
(102,109)(103,108)(104,107)(105,106)(127,166)(128,165)(129,164)(130,163)
(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)(138,155)
(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)(146,147)
(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)(175,200)
(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)(183,192)
(184,191)(185,190)(186,189)(187,188)(209,248)(210,247)(211,246)(212,245)
(213,244)(214,243)(215,242)(216,241)(217,240)(218,239)(219,238)(220,237)
(221,236)(222,235)(223,234)(224,233)(225,232)(226,231)(227,230)(228,229);;
s2 := (  3,  4)(  5, 43)(  6, 42)(  7, 41)(  8, 40)(  9, 39)( 10, 38)( 11, 37)
( 12, 36)( 13, 35)( 14, 34)( 15, 33)( 16, 32)( 17, 31)( 18, 30)( 19, 29)
( 20, 28)( 21, 27)( 22, 26)( 23, 25)( 44, 86)( 45, 85)( 46,125)( 47,124)
( 48,123)( 49,122)( 50,121)( 51,120)( 52,119)( 53,118)( 54,117)( 55,116)
( 56,115)( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)( 63,108)
( 64,107)( 65,106)( 66,105)( 67,104)( 68,103)( 69,102)( 70,101)( 71,100)
( 72, 99)( 73, 98)( 74, 97)( 75, 96)( 76, 95)( 77, 94)( 78, 93)( 79, 92)
( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)(126,127)(128,166)(129,165)
(130,164)(131,163)(132,162)(133,161)(134,160)(135,159)(136,158)(137,157)
(138,156)(139,155)(140,154)(141,153)(142,152)(143,151)(144,150)(145,149)
(146,148)(167,209)(168,208)(169,248)(170,247)(171,246)(172,245)(173,244)
(174,243)(175,242)(176,241)(177,240)(178,239)(179,238)(180,237)(181,236)
(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)(188,229)(189,228)
(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)(196,221)(197,220)
(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)(204,213)(205,212)
(206,211)(207,210);;
s3 := (  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)( 42,206)
( 43,207)( 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)
( 83,165)( 84,166)( 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)(124,247)(125,248);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(248)!(1,2);
s1 := Sym(248)!(  4, 43)(  5, 42)(  6, 41)(  7, 40)(  8, 39)(  9, 38)( 10, 37)
( 11, 36)( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16, 31)( 17, 30)( 18, 29)
( 19, 28)( 20, 27)( 21, 26)( 22, 25)( 23, 24)( 45, 84)( 46, 83)( 47, 82)
( 48, 81)( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)
( 56, 73)( 57, 72)( 58, 71)( 59, 70)( 60, 69)( 61, 68)( 62, 67)( 63, 66)
( 64, 65)( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)( 91,120)( 92,119)
( 93,118)( 94,117)( 95,116)( 96,115)( 97,114)( 98,113)( 99,112)(100,111)
(101,110)(102,109)(103,108)(104,107)(105,106)(127,166)(128,165)(129,164)
(130,163)(131,162)(132,161)(133,160)(134,159)(135,158)(136,157)(137,156)
(138,155)(139,154)(140,153)(141,152)(142,151)(143,150)(144,149)(145,148)
(146,147)(168,207)(169,206)(170,205)(171,204)(172,203)(173,202)(174,201)
(175,200)(176,199)(177,198)(178,197)(179,196)(180,195)(181,194)(182,193)
(183,192)(184,191)(185,190)(186,189)(187,188)(209,248)(210,247)(211,246)
(212,245)(213,244)(214,243)(215,242)(216,241)(217,240)(218,239)(219,238)
(220,237)(221,236)(222,235)(223,234)(224,233)(225,232)(226,231)(227,230)
(228,229);
s2 := Sym(248)!(  3,  4)(  5, 43)(  6, 42)(  7, 41)(  8, 40)(  9, 39)( 10, 38)
( 11, 37)( 12, 36)( 13, 35)( 14, 34)( 15, 33)( 16, 32)( 17, 31)( 18, 30)
( 19, 29)( 20, 28)( 21, 27)( 22, 26)( 23, 25)( 44, 86)( 45, 85)( 46,125)
( 47,124)( 48,123)( 49,122)( 50,121)( 51,120)( 52,119)( 53,118)( 54,117)
( 55,116)( 56,115)( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)
( 63,108)( 64,107)( 65,106)( 66,105)( 67,104)( 68,103)( 69,102)( 70,101)
( 71,100)( 72, 99)( 73, 98)( 74, 97)( 75, 96)( 76, 95)( 77, 94)( 78, 93)
( 79, 92)( 80, 91)( 81, 90)( 82, 89)( 83, 88)( 84, 87)(126,127)(128,166)
(129,165)(130,164)(131,163)(132,162)(133,161)(134,160)(135,159)(136,158)
(137,157)(138,156)(139,155)(140,154)(141,153)(142,152)(143,151)(144,150)
(145,149)(146,148)(167,209)(168,208)(169,248)(170,247)(171,246)(172,245)
(173,244)(174,243)(175,242)(176,241)(177,240)(178,239)(179,238)(180,237)
(181,236)(182,235)(183,234)(184,233)(185,232)(186,231)(187,230)(188,229)
(189,228)(190,227)(191,226)(192,225)(193,224)(194,223)(195,222)(196,221)
(197,220)(198,219)(199,218)(200,217)(201,216)(202,215)(203,214)(204,213)
(205,212)(206,211)(207,210);
s3 := Sym(248)!(  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)
( 42,206)( 43,207)( 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)( 83,165)( 84,166)( 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)(124,247)(125,248);
poly := sub<Sym(248)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope