Polytope of Type {110,4}

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

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