Polytope of Type {84,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1008b
Also Known As : {84,6|2}. if this polytope has another 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 : 2
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}*504b
   3-fold quotients : {28,6}*336a, {84,2}*336
   6-fold quotients : {14,6}*168, {42,2}*168
   7-fold quotients : {12,6}*144a
   9-fold quotients : {28,2}*112
   12-fold quotients : {21,2}*84
   14-fold quotients : {6,6}*72a
   18-fold quotients : {14,2}*56
   21-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {7,2}*28
   42-fold quotients : {2,6}*24, {6,2}*24
   63-fold quotients : {4,2}*16
   84-fold quotients : {2,3}*12, {3,2}*12
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

References : None.
to this polytope