Polytope of Type {4,63}

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

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