Polytope of Type {4,126}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,126}*1008a
Also Known As : {4,126|2}. if this polytope has another name.
Group : SmallGroup(1008,207)
Rank : 3
Schlafli Type : {4,126}
Number of vertices, edges, etc : 4, 252, 126
Order of s₀s₁s₂ : 252
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 : {2,126}*504
   3-fold quotients : {4,42}*336a
   4-fold quotients : {2,63}*252
   6-fold quotients : {2,42}*168
   7-fold quotients : {4,18}*144a
   9-fold quotients : {4,14}*112
   12-fold quotients : {2,21}*84
   14-fold quotients : {2,18}*72
   18-fold quotients : {2,14}*56
   21-fold quotients : {4,6}*48a
   28-fold quotients : {2,9}*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 := (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)(148,211)(149,212)(150,213)
(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)(158,221)
(159,222)(160,223)(161,224)(162,225)(163,226)(164,227)(165,228)(166,229)
(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)(174,237)
(175,238)(176,239)(177,240)(178,241)(179,242)(180,243)(181,244)(182,245)
(183,246)(184,247)(185,248)(186,249)(187,250)(188,251)(189,252);;
s1 := (  1,127)(  2,129)(  3,128)(  4,145)(  5,147)(  6,146)(  7,142)(  8,144)
(  9,143)( 10,139)( 11,141)( 12,140)( 13,136)( 14,138)( 15,137)( 16,133)
( 17,135)( 18,134)( 19,130)( 20,132)( 21,131)( 22,171)( 23,170)( 24,169)
( 25,189)( 26,188)( 27,187)( 28,186)( 29,185)( 30,184)( 31,183)( 32,182)
( 33,181)( 34,180)( 35,179)( 36,178)( 37,177)( 38,176)( 39,175)( 40,174)
( 41,173)( 42,172)( 43,150)( 44,149)( 45,148)( 46,168)( 47,167)( 48,166)
( 49,165)( 50,164)( 51,163)( 52,162)( 53,161)( 54,160)( 55,159)( 56,158)
( 57,157)( 58,156)( 59,155)( 60,154)( 61,153)( 62,152)( 63,151)( 64,190)
( 65,192)( 66,191)( 67,208)( 68,210)( 69,209)( 70,205)( 71,207)( 72,206)
( 73,202)( 74,204)( 75,203)( 76,199)( 77,201)( 78,200)( 79,196)( 80,198)
( 81,197)( 82,193)( 83,195)( 84,194)( 85,234)( 86,233)( 87,232)( 88,252)
( 89,251)( 90,250)( 91,249)( 92,248)( 93,247)( 94,246)( 95,245)( 96,244)
( 97,243)( 98,242)( 99,241)(100,240)(101,239)(102,238)(103,237)(104,236)
(105,235)(106,213)(107,212)(108,211)(109,231)(110,230)(111,229)(112,228)
(113,227)(114,226)(115,225)(116,224)(117,223)(118,222)(119,221)(120,220)
(121,219)(122,218)(123,217)(124,216)(125,215)(126,214);;
s2 := (  1, 25)(  2, 27)(  3, 26)(  4, 22)(  5, 24)(  6, 23)(  7, 40)(  8, 42)
(  9, 41)( 10, 37)( 11, 39)( 12, 38)( 13, 34)( 14, 36)( 15, 35)( 16, 31)
( 17, 33)( 18, 32)( 19, 28)( 20, 30)( 21, 29)( 43, 48)( 44, 47)( 45, 46)
( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)( 64, 88)
( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70,103)( 71,105)( 72,104)
( 73,100)( 74,102)( 75,101)( 76, 97)( 77, 99)( 78, 98)( 79, 94)( 80, 96)
( 81, 95)( 82, 91)( 83, 93)( 84, 92)(106,111)(107,110)(108,109)(112,126)
(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(127,151)(128,153)
(129,152)(130,148)(131,150)(132,149)(133,166)(134,168)(135,167)(136,163)
(137,165)(138,164)(139,160)(140,162)(141,161)(142,157)(143,159)(144,158)
(145,154)(146,156)(147,155)(169,174)(170,173)(171,172)(175,189)(176,188)
(177,187)(178,186)(179,185)(180,184)(181,183)(190,214)(191,216)(192,215)
(193,211)(194,213)(195,212)(196,229)(197,231)(198,230)(199,226)(200,228)
(201,227)(202,223)(203,225)(204,224)(205,220)(206,222)(207,221)(208,217)
(209,219)(210,218)(232,237)(233,236)(234,235)(238,252)(239,251)(240,250)
(241,249)(242,248)(243,247)(244,246);;
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*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*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)!(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)(148,211)(149,212)
(150,213)(151,214)(152,215)(153,216)(154,217)(155,218)(156,219)(157,220)
(158,221)(159,222)(160,223)(161,224)(162,225)(163,226)(164,227)(165,228)
(166,229)(167,230)(168,231)(169,232)(170,233)(171,234)(172,235)(173,236)
(174,237)(175,238)(176,239)(177,240)(178,241)(179,242)(180,243)(181,244)
(182,245)(183,246)(184,247)(185,248)(186,249)(187,250)(188,251)(189,252);
s1 := Sym(252)!(  1,127)(  2,129)(  3,128)(  4,145)(  5,147)(  6,146)(  7,142)
(  8,144)(  9,143)( 10,139)( 11,141)( 12,140)( 13,136)( 14,138)( 15,137)
( 16,133)( 17,135)( 18,134)( 19,130)( 20,132)( 21,131)( 22,171)( 23,170)
( 24,169)( 25,189)( 26,188)( 27,187)( 28,186)( 29,185)( 30,184)( 31,183)
( 32,182)( 33,181)( 34,180)( 35,179)( 36,178)( 37,177)( 38,176)( 39,175)
( 40,174)( 41,173)( 42,172)( 43,150)( 44,149)( 45,148)( 46,168)( 47,167)
( 48,166)( 49,165)( 50,164)( 51,163)( 52,162)( 53,161)( 54,160)( 55,159)
( 56,158)( 57,157)( 58,156)( 59,155)( 60,154)( 61,153)( 62,152)( 63,151)
( 64,190)( 65,192)( 66,191)( 67,208)( 68,210)( 69,209)( 70,205)( 71,207)
( 72,206)( 73,202)( 74,204)( 75,203)( 76,199)( 77,201)( 78,200)( 79,196)
( 80,198)( 81,197)( 82,193)( 83,195)( 84,194)( 85,234)( 86,233)( 87,232)
( 88,252)( 89,251)( 90,250)( 91,249)( 92,248)( 93,247)( 94,246)( 95,245)
( 96,244)( 97,243)( 98,242)( 99,241)(100,240)(101,239)(102,238)(103,237)
(104,236)(105,235)(106,213)(107,212)(108,211)(109,231)(110,230)(111,229)
(112,228)(113,227)(114,226)(115,225)(116,224)(117,223)(118,222)(119,221)
(120,220)(121,219)(122,218)(123,217)(124,216)(125,215)(126,214);
s2 := Sym(252)!(  1, 25)(  2, 27)(  3, 26)(  4, 22)(  5, 24)(  6, 23)(  7, 40)
(  8, 42)(  9, 41)( 10, 37)( 11, 39)( 12, 38)( 13, 34)( 14, 36)( 15, 35)
( 16, 31)( 17, 33)( 18, 32)( 19, 28)( 20, 30)( 21, 29)( 43, 48)( 44, 47)
( 45, 46)( 49, 63)( 50, 62)( 51, 61)( 52, 60)( 53, 59)( 54, 58)( 55, 57)
( 64, 88)( 65, 90)( 66, 89)( 67, 85)( 68, 87)( 69, 86)( 70,103)( 71,105)
( 72,104)( 73,100)( 74,102)( 75,101)( 76, 97)( 77, 99)( 78, 98)( 79, 94)
( 80, 96)( 81, 95)( 82, 91)( 83, 93)( 84, 92)(106,111)(107,110)(108,109)
(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(118,120)(127,151)
(128,153)(129,152)(130,148)(131,150)(132,149)(133,166)(134,168)(135,167)
(136,163)(137,165)(138,164)(139,160)(140,162)(141,161)(142,157)(143,159)
(144,158)(145,154)(146,156)(147,155)(169,174)(170,173)(171,172)(175,189)
(176,188)(177,187)(178,186)(179,185)(180,184)(181,183)(190,214)(191,216)
(192,215)(193,211)(194,213)(195,212)(196,229)(197,231)(198,230)(199,226)
(200,228)(201,227)(202,223)(203,225)(204,224)(205,220)(206,222)(207,221)
(208,217)(209,219)(210,218)(232,237)(233,236)(234,235)(238,252)(239,251)
(240,250)(241,249)(242,248)(243,247)(244,246);
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*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*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