Polytope of Type {6,84}

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

s0 := ( 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);;
s1 := (  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)(  8, 36)
(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 29)( 16, 35)
( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 44, 49)( 45, 48)( 46, 47)
( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)( 64, 85)
( 65, 91)( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 99)( 72,105)
( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)( 78, 92)( 79, 98)( 80, 97)
( 81, 96)( 82, 95)( 83, 94)( 84, 93)(107,112)(108,111)(109,110)(113,120)
(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,211)(128,217)
(129,216)(130,215)(131,214)(132,213)(133,212)(134,225)(135,231)(136,230)
(137,229)(138,228)(139,227)(140,226)(141,218)(142,224)(143,223)(144,222)
(145,221)(146,220)(147,219)(148,190)(149,196)(150,195)(151,194)(152,193)
(153,192)(154,191)(155,204)(156,210)(157,209)(158,208)(159,207)(160,206)
(161,205)(162,197)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)
(169,232)(170,238)(171,237)(172,236)(173,235)(174,234)(175,233)(176,246)
(177,252)(178,251)(179,250)(180,249)(181,248)(182,247)(183,239)(184,245)
(185,244)(186,243)(187,242)(188,241)(189,240);;
s2 := (  1,135)(  2,134)(  3,140)(  4,139)(  5,138)(  6,137)(  7,136)(  8,128)
(  9,127)( 10,133)( 11,132)( 12,131)( 13,130)( 14,129)( 15,142)( 16,141)
( 17,147)( 18,146)( 19,145)( 20,144)( 21,143)( 22,177)( 23,176)( 24,182)
( 25,181)( 26,180)( 27,179)( 28,178)( 29,170)( 30,169)( 31,175)( 32,174)
( 33,173)( 34,172)( 35,171)( 36,184)( 37,183)( 38,189)( 39,188)( 40,187)
( 41,186)( 42,185)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)( 48,158)
( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)( 56,150)
( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)( 64,198)
( 65,197)( 66,203)( 67,202)( 68,201)( 69,200)( 70,199)( 71,191)( 72,190)
( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,205)( 79,204)( 80,210)
( 81,209)( 82,208)( 83,207)( 84,206)( 85,240)( 86,239)( 87,245)( 88,244)
( 89,243)( 90,242)( 91,241)( 92,233)( 93,232)( 94,238)( 95,237)( 96,236)
( 97,235)( 98,234)( 99,247)(100,246)(101,252)(102,251)(103,250)(104,249)
(105,248)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)(112,220)
(113,212)(114,211)(115,217)(116,216)(117,215)(118,214)(119,213)(120,226)
(121,225)(122,231)(123,230)(124,229)(125,228)(126,227);;
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*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := 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);
s1 := Sym(252)!(  1, 22)(  2, 28)(  3, 27)(  4, 26)(  5, 25)(  6, 24)(  7, 23)
(  8, 36)(  9, 42)( 10, 41)( 11, 40)( 12, 39)( 13, 38)( 14, 37)( 15, 29)
( 16, 35)( 17, 34)( 18, 33)( 19, 32)( 20, 31)( 21, 30)( 44, 49)( 45, 48)
( 46, 47)( 50, 57)( 51, 63)( 52, 62)( 53, 61)( 54, 60)( 55, 59)( 56, 58)
( 64, 85)( 65, 91)( 66, 90)( 67, 89)( 68, 88)( 69, 87)( 70, 86)( 71, 99)
( 72,105)( 73,104)( 74,103)( 75,102)( 76,101)( 77,100)( 78, 92)( 79, 98)
( 80, 97)( 81, 96)( 82, 95)( 83, 94)( 84, 93)(107,112)(108,111)(109,110)
(113,120)(114,126)(115,125)(116,124)(117,123)(118,122)(119,121)(127,211)
(128,217)(129,216)(130,215)(131,214)(132,213)(133,212)(134,225)(135,231)
(136,230)(137,229)(138,228)(139,227)(140,226)(141,218)(142,224)(143,223)
(144,222)(145,221)(146,220)(147,219)(148,190)(149,196)(150,195)(151,194)
(152,193)(153,192)(154,191)(155,204)(156,210)(157,209)(158,208)(159,207)
(160,206)(161,205)(162,197)(163,203)(164,202)(165,201)(166,200)(167,199)
(168,198)(169,232)(170,238)(171,237)(172,236)(173,235)(174,234)(175,233)
(176,246)(177,252)(178,251)(179,250)(180,249)(181,248)(182,247)(183,239)
(184,245)(185,244)(186,243)(187,242)(188,241)(189,240);
s2 := Sym(252)!(  1,135)(  2,134)(  3,140)(  4,139)(  5,138)(  6,137)(  7,136)
(  8,128)(  9,127)( 10,133)( 11,132)( 12,131)( 13,130)( 14,129)( 15,142)
( 16,141)( 17,147)( 18,146)( 19,145)( 20,144)( 21,143)( 22,177)( 23,176)
( 24,182)( 25,181)( 26,180)( 27,179)( 28,178)( 29,170)( 30,169)( 31,175)
( 32,174)( 33,173)( 34,172)( 35,171)( 36,184)( 37,183)( 38,189)( 39,188)
( 40,187)( 41,186)( 42,185)( 43,156)( 44,155)( 45,161)( 46,160)( 47,159)
( 48,158)( 49,157)( 50,149)( 51,148)( 52,154)( 53,153)( 54,152)( 55,151)
( 56,150)( 57,163)( 58,162)( 59,168)( 60,167)( 61,166)( 62,165)( 63,164)
( 64,198)( 65,197)( 66,203)( 67,202)( 68,201)( 69,200)( 70,199)( 71,191)
( 72,190)( 73,196)( 74,195)( 75,194)( 76,193)( 77,192)( 78,205)( 79,204)
( 80,210)( 81,209)( 82,208)( 83,207)( 84,206)( 85,240)( 86,239)( 87,245)
( 88,244)( 89,243)( 90,242)( 91,241)( 92,233)( 93,232)( 94,238)( 95,237)
( 96,236)( 97,235)( 98,234)( 99,247)(100,246)(101,252)(102,251)(103,250)
(104,249)(105,248)(106,219)(107,218)(108,224)(109,223)(110,222)(111,221)
(112,220)(113,212)(114,211)(115,217)(116,216)(117,215)(118,214)(119,213)
(120,226)(121,225)(122,231)(123,230)(124,229)(125,228)(126,227);
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*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*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope