Polytope of Type {42,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,12}*1008b
Also Known As : {42,12|2}. if this polytope has another name.
Group : SmallGroup(1008,782)
Rank : 3
Schlafli Type : {42,12}
Number of vertices, edges, etc : 42, 252, 12
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 : {42,6}*504b
   3-fold quotients : {14,12}*336, {42,4}*336a
   6-fold quotients : {14,6}*168, {42,2}*168
   7-fold quotients : {6,12}*144a
   9-fold quotients : {14,4}*112
   12-fold quotients : {21,2}*84
   14-fold quotients : {6,6}*72a
   18-fold quotients : {14,2}*56
   21-fold quotients : {2,12}*48, {6,4}*48a
   36-fold quotients : {7,2}*28
   42-fold quotients : {2,6}*24, {6,2}*24
   63-fold quotients : {2,4}*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)(128,133)(129,132)(130,131)(134,141)
(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)(150,153)
(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)(161,163)
(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)(180,186)
(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)(199,209)
(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)(218,225)
(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)(234,237)
(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)(245,247);;
s1 := (  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)(127,198)(128,197)
(129,203)(130,202)(131,201)(132,200)(133,199)(134,191)(135,190)(136,196)
(137,195)(138,194)(139,193)(140,192)(141,205)(142,204)(143,210)(144,209)
(145,208)(146,207)(147,206)(148,240)(149,239)(150,245)(151,244)(152,243)
(153,242)(154,241)(155,233)(156,232)(157,238)(158,237)(159,236)(160,235)
(161,234)(162,247)(163,246)(164,252)(165,251)(166,250)(167,249)(168,248)
(169,219)(170,218)(171,224)(172,223)(173,222)(174,221)(175,220)(176,212)
(177,211)(178,217)(179,216)(180,215)(181,214)(182,213)(183,226)(184,225)
(185,231)(186,230)(187,229)(188,228)(189,227);;
s2 := (  1,148)(  2,149)(  3,150)(  4,151)(  5,152)(  6,153)(  7,154)(  8,155)
(  9,156)( 10,157)( 11,158)( 12,159)( 13,160)( 14,161)( 15,162)( 16,163)
( 17,164)( 18,165)( 19,166)( 20,167)( 21,168)( 22,127)( 23,128)( 24,129)
( 25,130)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,136)( 32,137)
( 33,138)( 34,139)( 35,140)( 36,141)( 37,142)( 38,143)( 39,144)( 40,145)
( 41,146)( 42,147)( 43,169)( 44,170)( 45,171)( 46,172)( 47,173)( 48,174)
( 49,175)( 50,176)( 51,177)( 52,178)( 53,179)( 54,180)( 55,181)( 56,182)
( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)( 64,211)
( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,218)( 72,219)
( 73,220)( 74,221)( 75,222)( 76,223)( 77,224)( 78,225)( 79,226)( 80,227)
( 81,228)( 82,229)( 83,230)( 84,231)( 85,190)( 86,191)( 87,192)( 88,193)
( 89,194)( 90,195)( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)( 96,201)
( 97,202)( 98,203)( 99,204)(100,205)(101,206)(102,207)(103,208)(104,209)
(105,210)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)(112,238)
(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)(120,246)
(121,247)(122,248)(123,249)(124,250)(125,251)(126,252);;
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, s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)(128,133)(129,132)(130,131)
(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)(149,154)
(150,153)(151,152)(155,162)(156,168)(157,167)(158,166)(159,165)(160,164)
(161,163)(170,175)(171,174)(172,173)(176,183)(177,189)(178,188)(179,187)
(180,186)(181,185)(182,184)(191,196)(192,195)(193,194)(197,204)(198,210)
(199,209)(200,208)(201,207)(202,206)(203,205)(212,217)(213,216)(214,215)
(218,225)(219,231)(220,230)(221,229)(222,228)(223,227)(224,226)(233,238)
(234,237)(235,236)(239,246)(240,252)(241,251)(242,250)(243,249)(244,248)
(245,247);
s1 := 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)(127,198)
(128,197)(129,203)(130,202)(131,201)(132,200)(133,199)(134,191)(135,190)
(136,196)(137,195)(138,194)(139,193)(140,192)(141,205)(142,204)(143,210)
(144,209)(145,208)(146,207)(147,206)(148,240)(149,239)(150,245)(151,244)
(152,243)(153,242)(154,241)(155,233)(156,232)(157,238)(158,237)(159,236)
(160,235)(161,234)(162,247)(163,246)(164,252)(165,251)(166,250)(167,249)
(168,248)(169,219)(170,218)(171,224)(172,223)(173,222)(174,221)(175,220)
(176,212)(177,211)(178,217)(179,216)(180,215)(181,214)(182,213)(183,226)
(184,225)(185,231)(186,230)(187,229)(188,228)(189,227);
s2 := Sym(252)!(  1,148)(  2,149)(  3,150)(  4,151)(  5,152)(  6,153)(  7,154)
(  8,155)(  9,156)( 10,157)( 11,158)( 12,159)( 13,160)( 14,161)( 15,162)
( 16,163)( 17,164)( 18,165)( 19,166)( 20,167)( 21,168)( 22,127)( 23,128)
( 24,129)( 25,130)( 26,131)( 27,132)( 28,133)( 29,134)( 30,135)( 31,136)
( 32,137)( 33,138)( 34,139)( 35,140)( 36,141)( 37,142)( 38,143)( 39,144)
( 40,145)( 41,146)( 42,147)( 43,169)( 44,170)( 45,171)( 46,172)( 47,173)
( 48,174)( 49,175)( 50,176)( 51,177)( 52,178)( 53,179)( 54,180)( 55,181)
( 56,182)( 57,183)( 58,184)( 59,185)( 60,186)( 61,187)( 62,188)( 63,189)
( 64,211)( 65,212)( 66,213)( 67,214)( 68,215)( 69,216)( 70,217)( 71,218)
( 72,219)( 73,220)( 74,221)( 75,222)( 76,223)( 77,224)( 78,225)( 79,226)
( 80,227)( 81,228)( 82,229)( 83,230)( 84,231)( 85,190)( 86,191)( 87,192)
( 88,193)( 89,194)( 90,195)( 91,196)( 92,197)( 93,198)( 94,199)( 95,200)
( 96,201)( 97,202)( 98,203)( 99,204)(100,205)(101,206)(102,207)(103,208)
(104,209)(105,210)(106,232)(107,233)(108,234)(109,235)(110,236)(111,237)
(112,238)(113,239)(114,240)(115,241)(116,242)(117,243)(118,244)(119,245)
(120,246)(121,247)(122,248)(123,249)(124,250)(125,251)(126,252);
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, s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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