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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}*1008c
if this polytope has a name.
Group : SmallGroup(1008,797)
Rank : 3
Schlafli Type : {12,42}
Number of vertices, edges, etc : 12, 252, 42
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 : {4,42}*336a
   4-fold quotients : {6,21}*252
   6-fold quotients : {2,42}*168
   7-fold quotients : {12,6}*144c
   9-fold quotients : {4,14}*112
   12-fold quotients : {2,21}*84
   14-fold quotients : {6,6}*72b
   18-fold quotients : {2,14}*56
   21-fold quotients : {4,6}*48a
   28-fold quotients : {6,3}*36
   36-fold quotients : {2,7}*28
   42-fold quotients : {2,6}*24
   63-fold quotients : {4,2}*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)(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, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)( 27, 53)
( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)( 35, 45)
( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)( 64, 72)
( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)( 80, 84)
( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)( 91,115)
( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)( 99,121)
(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)(128,134)
(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)(144,146)
(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)(155,170)
(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)(163,183)
(164,189)(165,188)(166,187)(167,186)(168,185)(190,198)(191,197)(192,203)
(193,202)(194,201)(195,200)(196,199)(204,205)(206,210)(207,209)(211,240)
(212,239)(213,245)(214,244)(215,243)(216,242)(217,241)(218,233)(219,232)
(220,238)(221,237)(222,236)(223,235)(224,234)(225,247)(226,246)(227,252)
(228,251)(229,250)(230,249)(231,248);;
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*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, 51)( 23, 50)( 24, 56)( 25, 55)( 26, 54)
( 27, 53)( 28, 52)( 29, 44)( 30, 43)( 31, 49)( 32, 48)( 33, 47)( 34, 46)
( 35, 45)( 36, 58)( 37, 57)( 38, 63)( 39, 62)( 40, 61)( 41, 60)( 42, 59)
( 64, 72)( 65, 71)( 66, 77)( 67, 76)( 68, 75)( 69, 74)( 70, 73)( 78, 79)
( 80, 84)( 81, 83)( 85,114)( 86,113)( 87,119)( 88,118)( 89,117)( 90,116)
( 91,115)( 92,107)( 93,106)( 94,112)( 95,111)( 96,110)( 97,109)( 98,108)
( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)(105,122)(127,135)
(128,134)(129,140)(130,139)(131,138)(132,137)(133,136)(141,142)(143,147)
(144,146)(148,177)(149,176)(150,182)(151,181)(152,180)(153,179)(154,178)
(155,170)(156,169)(157,175)(158,174)(159,173)(160,172)(161,171)(162,184)
(163,183)(164,189)(165,188)(166,187)(167,186)(168,185)(190,198)(191,197)
(192,203)(193,202)(194,201)(195,200)(196,199)(204,205)(206,210)(207,209)
(211,240)(212,239)(213,245)(214,244)(215,243)(216,242)(217,241)(218,233)
(219,232)(220,238)(221,237)(222,236)(223,235)(224,234)(225,247)(226,246)
(227,252)(228,251)(229,250)(230,249)(231,248);
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*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