Polytope of Type {4,110}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,110}*880
Also Known As : {4,110|2}. if this polytope has another name.
Group : SmallGroup(880,200)
Rank : 3
Schlafli Type : {4,110}
Number of vertices, edges, etc : 4, 220, 110
Order of s₀s₁s₂ : 220
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,110,2} of size 1760
Vertex Figure Of :
   {2,4,110} of size 1760
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,110}*440
   4-fold quotients : {2,55}*220
   5-fold quotients : {4,22}*176
   10-fold quotients : {2,22}*88
   11-fold quotients : {4,10}*80
   20-fold quotients : {2,11}*44
   22-fold quotients : {2,10}*40
   44-fold quotients : {2,5}*20
   55-fold quotients : {4,2}*16
   110-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,220}*1760, {8,110}*1760
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope