Polytope of Type {20,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,22}*880
Also Known As : {20,22|2}. if this polytope has another name.
Group : SmallGroup(880,160)
Rank : 3
Schlafli Type : {20,22}
Number of vertices, edges, etc : 20, 220, 22
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 :
   {20,22,2} of size 1760
Vertex Figure Of :
   {2,20,22} of size 1760
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,22}*440
   5-fold quotients : {4,22}*176
   10-fold quotients : {2,22}*88
   11-fold quotients : {20,2}*80
   20-fold quotients : {2,11}*44
   22-fold quotients : {10,2}*40
   44-fold quotients : {5,2}*20
   55-fold quotients : {4,2}*16
   110-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {40,22}*1760, {20,44}*1760
Permutation Representation (GAP) :

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