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Polytope of Type {10,22,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,22,4}*1760
Also Known As : {{10,22|2},{22,4|2}}. if this polytope has another name.
Group : SmallGroup(1760,1190)
Rank : 4
Schlafli Type : {10,22,4}
Number of vertices, edges, etc : 10, 110, 44, 4
Order of s0s1s2s3 : 220
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {10,22,2}*880
   5-fold quotients : {2,22,4}*352
   10-fold quotients : {2,22,2}*176
   11-fold quotients : {10,2,4}*160
   20-fold quotients : {2,11,2}*88
   22-fold quotients : {5,2,4}*80, {10,2,2}*80
   44-fold quotients : {5,2,2}*40
   55-fold quotients : {2,2,4}*32
   110-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)(122,155)(123,156)(124,157)(125,158)
(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165)(133,144)
(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)
(142,153)(143,154)(177,210)(178,211)(179,212)(180,213)(181,214)(182,215)
(183,216)(184,217)(185,218)(186,219)(187,220)(188,199)(189,200)(190,201)
(191,202)(192,203)(193,204)(194,205)(195,206)(196,207)(197,208)(198,209);;
s1 := (  1, 12)(  2, 22)(  3, 21)(  4, 20)(  5, 19)(  6, 18)(  7, 17)(  8, 16)
(  9, 15)( 10, 14)( 11, 13)( 23, 45)( 24, 55)( 25, 54)( 26, 53)( 27, 52)
( 28, 51)( 29, 50)( 30, 49)( 31, 48)( 32, 47)( 33, 46)( 35, 44)( 36, 43)
( 37, 42)( 38, 41)( 39, 40)( 56, 67)( 57, 77)( 58, 76)( 59, 75)( 60, 74)
( 61, 73)( 62, 72)( 63, 71)( 64, 70)( 65, 69)( 66, 68)( 78,100)( 79,110)
( 80,109)( 81,108)( 82,107)( 83,106)( 84,105)( 85,104)( 86,103)( 87,102)
( 88,101)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)(111,122)(112,132)
(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)(120,124)
(121,123)(133,155)(134,165)(135,164)(136,163)(137,162)(138,161)(139,160)
(140,159)(141,158)(142,157)(143,156)(145,154)(146,153)(147,152)(148,151)
(149,150)(166,177)(167,187)(168,186)(169,185)(170,184)(171,183)(172,182)
(173,181)(174,180)(175,179)(176,178)(188,210)(189,220)(190,219)(191,218)
(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(200,209)
(201,208)(202,207)(203,206)(204,205);;
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)(111,167)(112,166)(113,176)(114,175)(115,174)(116,173)
(117,172)(118,171)(119,170)(120,169)(121,168)(122,178)(123,177)(124,187)
(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)(132,179)
(133,189)(134,188)(135,198)(136,197)(137,196)(138,195)(139,194)(140,193)
(141,192)(142,191)(143,190)(144,200)(145,199)(146,209)(147,208)(148,207)
(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,211)(156,210)
(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)(164,213)
(165,212);;
s3 := (  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,s3]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
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)(122,155)(123,156)(124,157)
(125,158)(126,159)(127,160)(128,161)(129,162)(130,163)(131,164)(132,165)
(133,144)(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)
(141,152)(142,153)(143,154)(177,210)(178,211)(179,212)(180,213)(181,214)
(182,215)(183,216)(184,217)(185,218)(186,219)(187,220)(188,199)(189,200)
(190,201)(191,202)(192,203)(193,204)(194,205)(195,206)(196,207)(197,208)
(198,209);
s1 := Sym(220)!(  1, 12)(  2, 22)(  3, 21)(  4, 20)(  5, 19)(  6, 18)(  7, 17)
(  8, 16)(  9, 15)( 10, 14)( 11, 13)( 23, 45)( 24, 55)( 25, 54)( 26, 53)
( 27, 52)( 28, 51)( 29, 50)( 30, 49)( 31, 48)( 32, 47)( 33, 46)( 35, 44)
( 36, 43)( 37, 42)( 38, 41)( 39, 40)( 56, 67)( 57, 77)( 58, 76)( 59, 75)
( 60, 74)( 61, 73)( 62, 72)( 63, 71)( 64, 70)( 65, 69)( 66, 68)( 78,100)
( 79,110)( 80,109)( 81,108)( 82,107)( 83,106)( 84,105)( 85,104)( 86,103)
( 87,102)( 88,101)( 90, 99)( 91, 98)( 92, 97)( 93, 96)( 94, 95)(111,122)
(112,132)(113,131)(114,130)(115,129)(116,128)(117,127)(118,126)(119,125)
(120,124)(121,123)(133,155)(134,165)(135,164)(136,163)(137,162)(138,161)
(139,160)(140,159)(141,158)(142,157)(143,156)(145,154)(146,153)(147,152)
(148,151)(149,150)(166,177)(167,187)(168,186)(169,185)(170,184)(171,183)
(172,182)(173,181)(174,180)(175,179)(176,178)(188,210)(189,220)(190,219)
(191,218)(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)
(200,209)(201,208)(202,207)(203,206)(204,205);
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)(111,167)(112,166)(113,176)(114,175)(115,174)
(116,173)(117,172)(118,171)(119,170)(120,169)(121,168)(122,178)(123,177)
(124,187)(125,186)(126,185)(127,184)(128,183)(129,182)(130,181)(131,180)
(132,179)(133,189)(134,188)(135,198)(136,197)(137,196)(138,195)(139,194)
(140,193)(141,192)(142,191)(143,190)(144,200)(145,199)(146,209)(147,208)
(148,207)(149,206)(150,205)(151,204)(152,203)(153,202)(154,201)(155,211)
(156,210)(157,220)(158,219)(159,218)(160,217)(161,216)(162,215)(163,214)
(164,213)(165,212);
s3 := 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,s3>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, 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