Polytope of Type {84,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1008c
if this polytope has a name.
Group : SmallGroup(1008,783)
Rank : 3
Schlafli Type : {84,6}
Number of vertices, edges, etc : 84, 252, 6
Order of s0s1s2 : 84
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,6}*504c
   3-fold quotients : {84,2}*336
   4-fold quotients : {21,6}*252
   6-fold quotients : {42,2}*168
   7-fold quotients : {12,6}*144b
   9-fold quotients : {28,2}*112
   12-fold quotients : {21,2}*84
   14-fold quotients : {6,6}*72c
   18-fold quotients : {14,2}*56
   21-fold quotients : {12,2}*48
   28-fold quotients : {3,6}*36
   36-fold quotients : {7,2}*28
   42-fold quotients : {6,2}*24
   63-fold quotients : {4,2}*16
   84-fold quotients : {3,2}*12
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)( 13, 17)( 14, 16)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)( 27, 45)( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(127,190)(128,196)(129,195)(130,194)(131,193)(132,192)(133,191)(134,204)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(141,197)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)(148,232)(149,238)(150,237)(151,236)(152,235)(153,234)(154,233)(155,246)(156,252)(157,251)(158,250)(159,249)(160,248)(161,247)(162,239)(163,245)(164,244)(165,243)(166,242)(167,241)(168,240)(169,211)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,225)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,218)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219);;
s1 := (  1,156)(  2,155)(  3,161)(  4,160)(  5,159)(  6,158)(  7,157)(  8,149)(  9,148)( 10,154)( 11,153)( 12,152)( 13,151)( 14,150)( 15,163)( 16,162)( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,135)( 23,134)( 24,140)( 25,139)( 26,138)( 27,137)( 28,136)( 29,128)( 30,127)( 31,133)( 32,132)( 33,131)( 34,130)( 35,129)( 36,142)( 37,141)( 38,147)( 39,146)( 40,145)( 41,144)( 42,143)( 43,177)( 44,176)( 45,182)( 46,181)( 47,180)( 48,179)( 49,178)( 50,170)( 51,169)( 52,175)( 53,174)( 54,173)( 55,172)( 56,171)( 57,184)( 58,183)( 59,189)( 60,188)( 61,187)( 62,186)( 63,185)( 64,219)( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)( 80,231)( 81,230)( 82,229)( 83,228)( 84,227)( 85,198)( 86,197)( 87,203)( 88,202)( 89,201)( 90,200)( 91,199)( 92,191)( 93,190)( 94,196)( 95,195)( 96,194)( 97,193)( 98,192)( 99,205)(100,204)(101,210)(102,209)(103,208)(104,207)(105,206)(106,240)(107,239)(108,245)(109,244)(110,243)(111,242)(112,241)(113,233)(114,232)(115,238)(116,237)(117,236)(118,235)(119,234)(120,247)(121,246)(122,252)(123,251)(124,250)(125,249)(126,248);;
s2 := ( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237)(217,238)(218,239)(219,240)(220,241)(221,242)(222,243)(223,244)(224,245)(225,246)(226,247)(227,248)(228,249)(229,250)(230,251)(231,252);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(252)!(  2,  7)(  3,  6)(  4,  5)(  8, 15)(  9, 21)( 10, 20)( 11, 19)( 12, 18)( 13, 17)( 14, 16)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)( 27, 45)( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(127,190)(128,196)(129,195)(130,194)(131,193)(132,192)(133,191)(134,204)(135,210)(136,209)(137,208)(138,207)(139,206)(140,205)(141,197)(142,203)(143,202)(144,201)(145,200)(146,199)(147,198)(148,232)(149,238)(150,237)(151,236)(152,235)(153,234)(154,233)(155,246)(156,252)(157,251)(158,250)(159,249)(160,248)(161,247)(162,239)(163,245)(164,244)(165,243)(166,242)(167,241)(168,240)(169,211)(170,217)(171,216)(172,215)(173,214)(174,213)(175,212)(176,225)(177,231)(178,230)(179,229)(180,228)(181,227)(182,226)(183,218)(184,224)(185,223)(186,222)(187,221)(188,220)(189,219);
s1 := Sym(252)!(  1,156)(  2,155)(  3,161)(  4,160)(  5,159)(  6,158)(  7,157)(  8,149)(  9,148)( 10,154)( 11,153)( 12,152)( 13,151)( 14,150)( 15,163)( 16,162)( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,135)( 23,134)( 24,140)( 25,139)( 26,138)( 27,137)( 28,136)( 29,128)( 30,127)( 31,133)( 32,132)( 33,131)( 34,130)( 35,129)( 36,142)( 37,141)( 38,147)( 39,146)( 40,145)( 41,144)( 42,143)( 43,177)( 44,176)( 45,182)( 46,181)( 47,180)( 48,179)( 49,178)( 50,170)( 51,169)( 52,175)( 53,174)( 54,173)( 55,172)( 56,171)( 57,184)( 58,183)( 59,189)( 60,188)( 61,187)( 62,186)( 63,185)( 64,219)( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)( 80,231)( 81,230)( 82,229)( 83,228)( 84,227)( 85,198)( 86,197)( 87,203)( 88,202)( 89,201)( 90,200)( 91,199)( 92,191)( 93,190)( 94,196)( 95,195)( 96,194)( 97,193)( 98,192)( 99,205)(100,204)(101,210)(102,209)(103,208)(104,207)(105,206)(106,240)(107,239)(108,245)(109,244)(110,243)(111,242)(112,241)(113,233)(114,232)(115,238)(116,237)(117,236)(118,235)(119,234)(120,247)(121,246)(122,252)(123,251)(124,250)(125,249)(126,248);
s2 := Sym(252)!( 22, 43)( 23, 44)( 24, 45)( 25, 46)( 26, 47)( 27, 48)( 28, 49)( 29, 50)( 30, 51)( 31, 52)( 32, 53)( 33, 54)( 34, 55)( 35, 56)( 36, 57)( 37, 58)( 38, 59)( 39, 60)( 40, 61)( 41, 62)( 42, 63)( 85,106)( 86,107)( 87,108)( 88,109)( 89,110)( 90,111)( 91,112)( 92,113)( 93,114)( 94,115)( 95,116)( 96,117)( 97,118)( 98,119)( 99,120)(100,121)(101,122)(102,123)(103,124)(104,125)(105,126)(148,169)(149,170)(150,171)(151,172)(152,173)(153,174)(154,175)(155,176)(156,177)(157,178)(158,179)(159,180)(160,181)(161,182)(162,183)(163,184)(164,185)(165,186)(166,187)(167,188)(168,189)(211,232)(212,233)(213,234)(214,235)(215,236)(216,237)(217,238)(218,239)(219,240)(220,241)(221,242)(222,243)(223,244)(224,245)(225,246)(226,247)(227,248)(228,249)(229,250)(230,251)(231,252);
poly := sub<Sym(252)|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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*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*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle