Polytope of Type {42,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,12}*1008c
if this polytope has a name.
Group : SmallGroup(1008,797)
Rank : 3
Schlafli Type : {42,12}
Number of vertices, edges, etc : 42, 252, 12
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 : {42,6}*504c
   3-fold quotients : {42,4}*336a
   4-fold quotients : {21,6}*252
   6-fold quotients : {42,2}*168
   7-fold quotients : {6,12}*144c
   9-fold quotients : {14,4}*112
   12-fold quotients : {21,2}*84
   14-fold quotients : {6,6}*72c
   18-fold quotients : {14,2}*56
   21-fold quotients : {6,4}*48a
   28-fold quotients : {3,6}*36
   36-fold quotients : {7,2}*28
   42-fold quotients : {6,2}*24
   63-fold quotients : {2,4}*16
   84-fold quotients : {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)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)( 27, 45)
( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)( 35, 58)
( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)( 65, 70)
( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)( 76, 80)
( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)( 91,107)
( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)( 99,113)
(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)(129,132)
(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)(140,142)
(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)(155,183)
(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)(163,182)
(164,181)(165,180)(166,179)(167,178)(168,177)(191,196)(192,195)(193,194)
(197,204)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)(211,232)
(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,246)(219,252)
(220,251)(221,250)(222,249)(223,248)(224,247)(225,239)(226,245)(227,244)
(228,243)(229,242)(230,241)(231,240);;
s1 := (  1,156)(  2,155)(  3,161)(  4,160)(  5,159)(  6,158)(  7,157)(  8,149)
(  9,148)( 10,154)( 11,153)( 12,152)( 13,151)( 14,150)( 15,163)( 16,162)
( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,135)( 23,134)( 24,140)
( 25,139)( 26,138)( 27,137)( 28,136)( 29,128)( 30,127)( 31,133)( 32,132)
( 33,131)( 34,130)( 35,129)( 36,142)( 37,141)( 38,147)( 39,146)( 40,145)
( 41,144)( 42,143)( 43,177)( 44,176)( 45,182)( 46,181)( 47,180)( 48,179)
( 49,178)( 50,170)( 51,169)( 52,175)( 53,174)( 54,173)( 55,172)( 56,171)
( 57,184)( 58,183)( 59,189)( 60,188)( 61,187)( 62,186)( 63,185)( 64,219)
( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)( 72,211)
( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)( 80,231)
( 81,230)( 82,229)( 83,228)( 84,227)( 85,198)( 86,197)( 87,203)( 88,202)
( 89,201)( 90,200)( 91,199)( 92,191)( 93,190)( 94,196)( 95,195)( 96,194)
( 97,193)( 98,192)( 99,205)(100,204)(101,210)(102,209)(103,208)(104,207)
(105,206)(106,240)(107,239)(108,245)(109,244)(110,243)(111,242)(112,241)
(113,233)(114,232)(115,238)(116,237)(117,236)(118,235)(119,234)(120,247)
(121,246)(122,252)(123,251)(124,250)(125,249)(126,248);;
s2 := ( 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);;
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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)( 22, 43)( 23, 49)( 24, 48)( 25, 47)( 26, 46)
( 27, 45)( 28, 44)( 29, 57)( 30, 63)( 31, 62)( 32, 61)( 33, 60)( 34, 59)
( 35, 58)( 36, 50)( 37, 56)( 38, 55)( 39, 54)( 40, 53)( 41, 52)( 42, 51)
( 65, 70)( 66, 69)( 67, 68)( 71, 78)( 72, 84)( 73, 83)( 74, 82)( 75, 81)
( 76, 80)( 77, 79)( 85,106)( 86,112)( 87,111)( 88,110)( 89,109)( 90,108)
( 91,107)( 92,120)( 93,126)( 94,125)( 95,124)( 96,123)( 97,122)( 98,121)
( 99,113)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(128,133)
(129,132)(130,131)(134,141)(135,147)(136,146)(137,145)(138,144)(139,143)
(140,142)(148,169)(149,175)(150,174)(151,173)(152,172)(153,171)(154,170)
(155,183)(156,189)(157,188)(158,187)(159,186)(160,185)(161,184)(162,176)
(163,182)(164,181)(165,180)(166,179)(167,178)(168,177)(191,196)(192,195)
(193,194)(197,204)(198,210)(199,209)(200,208)(201,207)(202,206)(203,205)
(211,232)(212,238)(213,237)(214,236)(215,235)(216,234)(217,233)(218,246)
(219,252)(220,251)(221,250)(222,249)(223,248)(224,247)(225,239)(226,245)
(227,244)(228,243)(229,242)(230,241)(231,240);
s1 := Sym(252)!(  1,156)(  2,155)(  3,161)(  4,160)(  5,159)(  6,158)(  7,157)
(  8,149)(  9,148)( 10,154)( 11,153)( 12,152)( 13,151)( 14,150)( 15,163)
( 16,162)( 17,168)( 18,167)( 19,166)( 20,165)( 21,164)( 22,135)( 23,134)
( 24,140)( 25,139)( 26,138)( 27,137)( 28,136)( 29,128)( 30,127)( 31,133)
( 32,132)( 33,131)( 34,130)( 35,129)( 36,142)( 37,141)( 38,147)( 39,146)
( 40,145)( 41,144)( 42,143)( 43,177)( 44,176)( 45,182)( 46,181)( 47,180)
( 48,179)( 49,178)( 50,170)( 51,169)( 52,175)( 53,174)( 54,173)( 55,172)
( 56,171)( 57,184)( 58,183)( 59,189)( 60,188)( 61,187)( 62,186)( 63,185)
( 64,219)( 65,218)( 66,224)( 67,223)( 68,222)( 69,221)( 70,220)( 71,212)
( 72,211)( 73,217)( 74,216)( 75,215)( 76,214)( 77,213)( 78,226)( 79,225)
( 80,231)( 81,230)( 82,229)( 83,228)( 84,227)( 85,198)( 86,197)( 87,203)
( 88,202)( 89,201)( 90,200)( 91,199)( 92,191)( 93,190)( 94,196)( 95,195)
( 96,194)( 97,193)( 98,192)( 99,205)(100,204)(101,210)(102,209)(103,208)
(104,207)(105,206)(106,240)(107,239)(108,245)(109,244)(110,243)(111,242)
(112,241)(113,233)(114,232)(115,238)(116,237)(117,236)(118,235)(119,234)
(120,247)(121,246)(122,252)(123,251)(124,250)(125,249)(126,248);
s2 := 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);
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*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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