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Polytope of Type {12,42}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,42}*1008b
Also Known As : {12,42|2}. if this polytope has another name.
Group : SmallGroup(1008,782)
Rank : 3
Schlafli Type : {12,42}
Number of vertices, edges, etc : 12, 252, 42
Order of s0s1s2 : 84
Order of s0s1s2s1 : 2
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}*504b
   3-fold quotients : {12,14}*336, {4,42}*336a
   6-fold quotients : {6,14}*168, {2,42}*168
   7-fold quotients : {12,6}*144a
   9-fold quotients : {4,14}*112
   12-fold quotients : {2,21}*84
   14-fold quotients : {6,6}*72a
   18-fold quotients : {2,14}*56
   21-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {2,7}*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 := ( 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)(127,190)(128,191)(129,192)(130,193)(131,194)(132,195)
(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)(140,203)
(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)(148,232)
(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)(156,240)
(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)(164,248)
(165,249)(166,250)(167,251)(168,252)(169,211)(170,212)(171,213)(172,214)
(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)(180,222)
(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)(188,230)
(189,231);;
s1 := (  1,148)(  2,154)(  3,153)(  4,152)(  5,151)(  6,150)(  7,149)(  8,162)
(  9,168)( 10,167)( 11,166)( 12,165)( 13,164)( 14,163)( 15,155)( 16,161)
( 17,160)( 18,159)( 19,158)( 20,157)( 21,156)( 22,127)( 23,133)( 24,132)
( 25,131)( 26,130)( 27,129)( 28,128)( 29,141)( 30,147)( 31,146)( 32,145)
( 33,144)( 34,143)( 35,142)( 36,134)( 37,140)( 38,139)( 39,138)( 40,137)
( 41,136)( 42,135)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)( 48,171)
( 49,170)( 50,183)( 51,189)( 52,188)( 53,187)( 54,186)( 55,185)( 56,184)
( 57,176)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)( 64,211)
( 65,217)( 66,216)( 67,215)( 68,214)( 69,213)( 70,212)( 71,225)( 72,231)
( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,218)( 79,224)( 80,223)
( 81,222)( 82,221)( 83,220)( 84,219)( 85,190)( 86,196)( 87,195)( 88,194)
( 89,193)( 90,192)( 91,191)( 92,204)( 93,210)( 94,209)( 95,208)( 96,207)
( 97,206)( 98,205)( 99,197)(100,203)(101,202)(102,201)(103,200)(104,199)
(105,198)(106,232)(107,238)(108,237)(109,236)(110,235)(111,234)(112,233)
(113,246)(114,252)(115,251)(116,250)(117,249)(118,248)(119,247)(120,239)
(121,245)(122,244)(123,243)(124,242)(125,241)(126,240);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 16)
( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)( 27, 32)
( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)( 46, 55)
( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 72)( 65, 71)
( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)( 81, 83)
( 85, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)( 99,100)
(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)(111,116)
(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)(130,139)
(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)(149,155)
(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)(165,167)
(169,177)(170,176)(171,182)(172,181)(173,180)(174,179)(175,178)(183,184)
(185,189)(186,188)(190,198)(191,197)(192,203)(193,202)(194,201)(195,200)
(196,199)(204,205)(206,210)(207,209)(211,219)(212,218)(213,224)(214,223)
(215,222)(216,221)(217,220)(225,226)(227,231)(228,230)(232,240)(233,239)
(234,245)(235,244)(236,243)(237,242)(238,241)(246,247)(248,252)(249,251);;
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, 
s0*s1*s0*s1*s0*s1*s0*s1*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 ];;
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)(127,190)(128,191)(129,192)(130,193)(131,194)
(132,195)(133,196)(134,197)(135,198)(136,199)(137,200)(138,201)(139,202)
(140,203)(141,204)(142,205)(143,206)(144,207)(145,208)(146,209)(147,210)
(148,232)(149,233)(150,234)(151,235)(152,236)(153,237)(154,238)(155,239)
(156,240)(157,241)(158,242)(159,243)(160,244)(161,245)(162,246)(163,247)
(164,248)(165,249)(166,250)(167,251)(168,252)(169,211)(170,212)(171,213)
(172,214)(173,215)(174,216)(175,217)(176,218)(177,219)(178,220)(179,221)
(180,222)(181,223)(182,224)(183,225)(184,226)(185,227)(186,228)(187,229)
(188,230)(189,231);
s1 := Sym(252)!(  1,148)(  2,154)(  3,153)(  4,152)(  5,151)(  6,150)(  7,149)
(  8,162)(  9,168)( 10,167)( 11,166)( 12,165)( 13,164)( 14,163)( 15,155)
( 16,161)( 17,160)( 18,159)( 19,158)( 20,157)( 21,156)( 22,127)( 23,133)
( 24,132)( 25,131)( 26,130)( 27,129)( 28,128)( 29,141)( 30,147)( 31,146)
( 32,145)( 33,144)( 34,143)( 35,142)( 36,134)( 37,140)( 38,139)( 39,138)
( 40,137)( 41,136)( 42,135)( 43,169)( 44,175)( 45,174)( 46,173)( 47,172)
( 48,171)( 49,170)( 50,183)( 51,189)( 52,188)( 53,187)( 54,186)( 55,185)
( 56,184)( 57,176)( 58,182)( 59,181)( 60,180)( 61,179)( 62,178)( 63,177)
( 64,211)( 65,217)( 66,216)( 67,215)( 68,214)( 69,213)( 70,212)( 71,225)
( 72,231)( 73,230)( 74,229)( 75,228)( 76,227)( 77,226)( 78,218)( 79,224)
( 80,223)( 81,222)( 82,221)( 83,220)( 84,219)( 85,190)( 86,196)( 87,195)
( 88,194)( 89,193)( 90,192)( 91,191)( 92,204)( 93,210)( 94,209)( 95,208)
( 96,207)( 97,206)( 98,205)( 99,197)(100,203)(101,202)(102,201)(103,200)
(104,199)(105,198)(106,232)(107,238)(108,237)(109,236)(110,235)(111,234)
(112,233)(113,246)(114,252)(115,251)(116,250)(117,249)(118,248)(119,247)
(120,239)(121,245)(122,244)(123,243)(124,242)(125,241)(126,240);
s2 := Sym(252)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 16)( 17, 21)( 18, 20)( 22, 30)( 23, 29)( 24, 35)( 25, 34)( 26, 33)
( 27, 32)( 28, 31)( 36, 37)( 38, 42)( 39, 41)( 43, 51)( 44, 50)( 45, 56)
( 46, 55)( 47, 54)( 48, 53)( 49, 52)( 57, 58)( 59, 63)( 60, 62)( 64, 72)
( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)
( 81, 83)( 85, 93)( 86, 92)( 87, 98)( 88, 97)( 89, 96)( 90, 95)( 91, 94)
( 99,100)(101,105)(102,104)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,121)(122,126)(123,125)(127,135)(128,134)(129,140)
(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)(148,156)
(149,155)(150,161)(151,160)(152,159)(153,158)(154,157)(162,163)(164,168)
(165,167)(169,177)(170,176)(171,182)(172,181)(173,180)(174,179)(175,178)
(183,184)(185,189)(186,188)(190,198)(191,197)(192,203)(193,202)(194,201)
(195,200)(196,199)(204,205)(206,210)(207,209)(211,219)(212,218)(213,224)
(214,223)(215,222)(216,221)(217,220)(225,226)(227,231)(228,230)(232,240)
(233,239)(234,245)(235,244)(236,243)(237,242)(238,241)(246,247)(248,252)
(249,251);
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, 
s0*s1*s0*s1*s0*s1*s0*s1*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 >; 
 
References : None.
to this polytope