Polytope of Type {4,106}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,106}*848
Also Known As : {4,106|2}. if this polytope has another name.
Group : SmallGroup(848,39)
Rank : 3
Schlafli Type : {4,106}
Number of vertices, edges, etc : 4, 212, 106
Order of s₀s₁s₂ : 212
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,106,2} of size 1696
Vertex Figure Of :
   {2,4,106} of size 1696
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,106}*424
   4-fold quotients : {2,53}*212
   53-fold quotients : {4,2}*16
   106-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,212}*1696, {8,106}*1696
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope