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Polytope of Type {22,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {22,20}*880
Also Known As : {22,20|2}. if this polytope has another name.
Group : SmallGroup(880,160)
Rank : 3
Schlafli Type : {22,20}
Number of vertices, edges, etc : 22, 220, 20
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 :
   {22,20,2} of size 1760
Vertex Figure Of :
   {2,22,20} of size 1760
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {22,10}*440
   5-fold quotients : {22,4}*176
   10-fold quotients : {22,2}*88
   11-fold quotients : {2,20}*80
   20-fold quotients : {11,2}*44
   22-fold quotients : {2,10}*40
   44-fold quotients : {2,5}*20
   55-fold quotients : {2,4}*16
   110-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {22,40}*1760, {44,20}*1760
Permutation Representation (GAP) :
s0 := (  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 13, 22)( 14, 21)( 15, 20)
( 16, 19)( 17, 18)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 35, 44)
( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 46, 55)( 47, 54)( 48, 53)( 49, 52)
( 50, 51)( 57, 66)( 58, 65)( 59, 64)( 60, 63)( 61, 62)( 68, 77)( 69, 76)
( 70, 75)( 71, 74)( 72, 73)( 79, 88)( 80, 87)( 81, 86)( 82, 85)( 83, 84)
( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)(101,110)(102,109)(103,108)
(104,107)(105,106)(112,121)(113,120)(114,119)(115,118)(116,117)(123,132)
(124,131)(125,130)(126,129)(127,128)(134,143)(135,142)(136,141)(137,140)
(138,139)(145,154)(146,153)(147,152)(148,151)(149,150)(156,165)(157,164)
(158,163)(159,162)(160,161)(167,176)(168,175)(169,174)(170,173)(171,172)
(178,187)(179,186)(180,185)(181,184)(182,183)(189,198)(190,197)(191,196)
(192,195)(193,194)(200,209)(201,208)(202,207)(203,206)(204,205)(211,220)
(212,219)(213,218)(214,217)(215,216);;
s1 := (  1,  2)(  3, 11)(  4, 10)(  5,  9)(  6,  8)( 12, 46)( 13, 45)( 14, 55)
( 15, 54)( 16, 53)( 17, 52)( 18, 51)( 19, 50)( 20, 49)( 21, 48)( 22, 47)
( 23, 35)( 24, 34)( 25, 44)( 26, 43)( 27, 42)( 28, 41)( 29, 40)( 30, 39)
( 31, 38)( 32, 37)( 33, 36)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)
( 67,101)( 68,100)( 69,110)( 70,109)( 71,108)( 72,107)( 73,106)( 74,105)
( 75,104)( 76,103)( 77,102)( 78, 90)( 79, 89)( 80, 99)( 81, 98)( 82, 97)
( 83, 96)( 84, 95)( 85, 94)( 86, 93)( 87, 92)( 88, 91)(111,167)(112,166)
(113,176)(114,175)(115,174)(116,173)(117,172)(118,171)(119,170)(120,169)
(121,168)(122,211)(123,210)(124,220)(125,219)(126,218)(127,217)(128,216)
(129,215)(130,214)(131,213)(132,212)(133,200)(134,199)(135,209)(136,208)
(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)(144,189)
(145,188)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)(152,192)
(153,191)(154,190)(155,178)(156,177)(157,187)(158,186)(159,185)(160,184)
(161,183)(162,182)(163,181)(164,180)(165,179);;
s2 := (  1,122)(  2,123)(  3,124)(  4,125)(  5,126)(  6,127)(  7,128)(  8,129)
(  9,130)( 10,131)( 11,132)( 12,111)( 13,112)( 14,113)( 15,114)( 16,115)
( 17,116)( 18,117)( 19,118)( 20,119)( 21,120)( 22,121)( 23,155)( 24,156)
( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)( 32,164)
( 33,165)( 34,144)( 35,145)( 36,146)( 37,147)( 38,148)( 39,149)( 40,150)
( 41,151)( 42,152)( 43,153)( 44,154)( 45,133)( 46,134)( 47,135)( 48,136)
( 49,137)( 50,138)( 51,139)( 52,140)( 53,141)( 54,142)( 55,143)( 56,177)
( 57,178)( 58,179)( 59,180)( 60,181)( 61,182)( 62,183)( 63,184)( 64,185)
( 65,186)( 66,187)( 67,166)( 68,167)( 69,168)( 70,169)( 71,170)( 72,171)
( 73,172)( 74,173)( 75,174)( 76,175)( 77,176)( 78,210)( 79,211)( 80,212)
( 81,213)( 82,214)( 83,215)( 84,216)( 85,217)( 86,218)( 87,219)( 88,220)
( 89,199)( 90,200)( 91,201)( 92,202)( 93,203)( 94,204)( 95,205)( 96,206)
( 97,207)( 98,208)( 99,209)(100,188)(101,189)(102,190)(103,191)(104,192)
(105,193)(106,194)(107,195)(108,196)(109,197)(110,198);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(220)!(  2, 11)(  3, 10)(  4,  9)(  5,  8)(  6,  7)( 13, 22)( 14, 21)
( 15, 20)( 16, 19)( 17, 18)( 24, 33)( 25, 32)( 26, 31)( 27, 30)( 28, 29)
( 35, 44)( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 46, 55)( 47, 54)( 48, 53)
( 49, 52)( 50, 51)( 57, 66)( 58, 65)( 59, 64)( 60, 63)( 61, 62)( 68, 77)
( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 79, 88)( 80, 87)( 81, 86)( 82, 85)
( 83, 84)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)(101,110)(102,109)
(103,108)(104,107)(105,106)(112,121)(113,120)(114,119)(115,118)(116,117)
(123,132)(124,131)(125,130)(126,129)(127,128)(134,143)(135,142)(136,141)
(137,140)(138,139)(145,154)(146,153)(147,152)(148,151)(149,150)(156,165)
(157,164)(158,163)(159,162)(160,161)(167,176)(168,175)(169,174)(170,173)
(171,172)(178,187)(179,186)(180,185)(181,184)(182,183)(189,198)(190,197)
(191,196)(192,195)(193,194)(200,209)(201,208)(202,207)(203,206)(204,205)
(211,220)(212,219)(213,218)(214,217)(215,216);
s1 := Sym(220)!(  1,  2)(  3, 11)(  4, 10)(  5,  9)(  6,  8)( 12, 46)( 13, 45)
( 14, 55)( 15, 54)( 16, 53)( 17, 52)( 18, 51)( 19, 50)( 20, 49)( 21, 48)
( 22, 47)( 23, 35)( 24, 34)( 25, 44)( 26, 43)( 27, 42)( 28, 41)( 29, 40)
( 30, 39)( 31, 38)( 32, 37)( 33, 36)( 56, 57)( 58, 66)( 59, 65)( 60, 64)
( 61, 63)( 67,101)( 68,100)( 69,110)( 70,109)( 71,108)( 72,107)( 73,106)
( 74,105)( 75,104)( 76,103)( 77,102)( 78, 90)( 79, 89)( 80, 99)( 81, 98)
( 82, 97)( 83, 96)( 84, 95)( 85, 94)( 86, 93)( 87, 92)( 88, 91)(111,167)
(112,166)(113,176)(114,175)(115,174)(116,173)(117,172)(118,171)(119,170)
(120,169)(121,168)(122,211)(123,210)(124,220)(125,219)(126,218)(127,217)
(128,216)(129,215)(130,214)(131,213)(132,212)(133,200)(134,199)(135,209)
(136,208)(137,207)(138,206)(139,205)(140,204)(141,203)(142,202)(143,201)
(144,189)(145,188)(146,198)(147,197)(148,196)(149,195)(150,194)(151,193)
(152,192)(153,191)(154,190)(155,178)(156,177)(157,187)(158,186)(159,185)
(160,184)(161,183)(162,182)(163,181)(164,180)(165,179);
s2 := Sym(220)!(  1,122)(  2,123)(  3,124)(  4,125)(  5,126)(  6,127)(  7,128)
(  8,129)(  9,130)( 10,131)( 11,132)( 12,111)( 13,112)( 14,113)( 15,114)
( 16,115)( 17,116)( 18,117)( 19,118)( 20,119)( 21,120)( 22,121)( 23,155)
( 24,156)( 25,157)( 26,158)( 27,159)( 28,160)( 29,161)( 30,162)( 31,163)
( 32,164)( 33,165)( 34,144)( 35,145)( 36,146)( 37,147)( 38,148)( 39,149)
( 40,150)( 41,151)( 42,152)( 43,153)( 44,154)( 45,133)( 46,134)( 47,135)
( 48,136)( 49,137)( 50,138)( 51,139)( 52,140)( 53,141)( 54,142)( 55,143)
( 56,177)( 57,178)( 58,179)( 59,180)( 60,181)( 61,182)( 62,183)( 63,184)
( 64,185)( 65,186)( 66,187)( 67,166)( 68,167)( 69,168)( 70,169)( 71,170)
( 72,171)( 73,172)( 74,173)( 75,174)( 76,175)( 77,176)( 78,210)( 79,211)
( 80,212)( 81,213)( 82,214)( 83,215)( 84,216)( 85,217)( 86,218)( 87,219)
( 88,220)( 89,199)( 90,200)( 91,201)( 92,202)( 93,203)( 94,204)( 95,205)
( 96,206)( 97,207)( 98,208)( 99,209)(100,188)(101,189)(102,190)(103,191)
(104,192)(105,193)(106,194)(107,195)(108,196)(109,197)(110,198);
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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 >; 
 
References : None.
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