Polytope of Type {63,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {63,4}*504
if this polytope has a name.
Group : SmallGroup(504,55)
Rank : 3
Schlafli Type : {63,4}
Number of vertices, edges, etc : 63, 126, 4
Order of s₀s₁s₂ : 63
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {63,4,2} of size 1008
Vertex Figure Of :
   {2,63,4} of size 1008
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {21,4}*168
   7-fold quotients : {9,4}*72
   21-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {63,4}*1008, {126,4}*1008b, {126,4}*1008c
   3-fold covers : {189,4}*1512
Permutation Representation (GAP) :

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