Polytope of Type {4,22,10}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,22,10}*1760
Also Known As : {{4,22|2},{22,10|2}}. if this polytope has another name.
Group : SmallGroup(1760,1190)
Rank : 4
Schlafli Type : {4,22,10}
Number of vertices, edges, etc : 4, 44, 110, 10
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 : {2,22,10}*880
5-fold quotients : {4,22,2}*352
10-fold quotients : {2,22,2}*176
11-fold quotients : {4,2,10}*160
20-fold quotients : {2,11,2}*88
22-fold quotients : {4,2,5}*80, {2,2,10}*80
44-fold quotients : {2,2,5}*40
55-fold quotients : {4,2,2}*32
110-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
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)( 12,122)( 13,132)( 14,131)( 15,130)( 16,129)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,133)( 24,143)
( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)( 32,135)
( 33,134)( 34,144)( 35,154)( 36,153)( 37,152)( 38,151)( 39,150)( 40,149)
( 41,148)( 42,147)( 43,146)( 44,145)( 45,155)( 46,165)( 47,164)( 48,163)
( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,156)( 56,166)
( 57,176)( 58,175)( 59,174)( 60,173)( 61,172)( 62,171)( 63,170)( 64,169)
( 65,168)( 66,167)( 67,177)( 68,187)( 69,186)( 70,185)( 71,184)( 72,183)
( 73,182)( 74,181)( 75,180)( 76,179)( 77,178)( 78,188)( 79,198)( 80,197)
( 81,196)( 82,195)( 83,194)( 84,193)( 85,192)( 86,191)( 87,190)( 88,189)
( 89,199)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)( 96,203)
( 97,202)( 98,201)( 99,200)(100,210)(101,220)(102,219)(103,218)(104,217)
(105,216)(106,215)(107,214)(108,213)(109,212)(110,211);;
s2 := ( 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)(111,112)(113,121)
(114,120)(115,119)(116,118)(122,156)(123,155)(124,165)(125,164)(126,163)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,145)(134,144)
(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)(142,147)
(143,146)(166,167)(168,176)(169,175)(170,174)(171,173)(177,211)(178,210)
(179,220)(180,219)(181,218)(182,217)(183,216)(184,215)(185,214)(186,213)
(187,212)(188,200)(189,199)(190,209)(191,208)(192,207)(193,206)(194,205)
(195,204)(196,203)(197,202)(198,201);;
s3 := ( 1, 12)( 2, 13)( 3, 14)( 4, 15)( 5, 16)( 6, 17)( 7, 18)( 8, 19)
( 9, 20)( 10, 21)( 11, 22)( 23, 45)( 24, 46)( 25, 47)( 26, 48)( 27, 49)
( 28, 50)( 29, 51)( 30, 52)( 31, 53)( 32, 54)( 33, 55)( 56, 67)( 57, 68)
( 58, 69)( 59, 70)( 60, 71)( 61, 72)( 62, 73)( 63, 74)( 64, 75)( 65, 76)
( 66, 77)( 78,100)( 79,101)( 80,102)( 81,103)( 82,104)( 83,105)( 84,106)
( 85,107)( 86,108)( 87,109)( 88,110)(111,122)(112,123)(113,124)(114,125)
(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)(133,155)
(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)(141,163)
(142,164)(143,165)(166,177)(167,178)(168,179)(169,180)(170,181)(171,182)
(172,183)(173,184)(174,185)(175,186)(176,187)(188,210)(189,211)(190,212)
(191,213)(192,214)(193,215)(194,216)(195,217)(196,218)(197,219)(198,220);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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)( 12,122)( 13,132)( 14,131)( 15,130)
( 16,129)( 17,128)( 18,127)( 19,126)( 20,125)( 21,124)( 22,123)( 23,133)
( 24,143)( 25,142)( 26,141)( 27,140)( 28,139)( 29,138)( 30,137)( 31,136)
( 32,135)( 33,134)( 34,144)( 35,154)( 36,153)( 37,152)( 38,151)( 39,150)
( 40,149)( 41,148)( 42,147)( 43,146)( 44,145)( 45,155)( 46,165)( 47,164)
( 48,163)( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,156)
( 56,166)( 57,176)( 58,175)( 59,174)( 60,173)( 61,172)( 62,171)( 63,170)
( 64,169)( 65,168)( 66,167)( 67,177)( 68,187)( 69,186)( 70,185)( 71,184)
( 72,183)( 73,182)( 74,181)( 75,180)( 76,179)( 77,178)( 78,188)( 79,198)
( 80,197)( 81,196)( 82,195)( 83,194)( 84,193)( 85,192)( 86,191)( 87,190)
( 88,189)( 89,199)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)
( 96,203)( 97,202)( 98,201)( 99,200)(100,210)(101,220)(102,219)(103,218)
(104,217)(105,216)(106,215)(107,214)(108,213)(109,212)(110,211);
s2 := 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)(111,112)
(113,121)(114,120)(115,119)(116,118)(122,156)(123,155)(124,165)(125,164)
(126,163)(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,145)
(134,144)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)(141,148)
(142,147)(143,146)(166,167)(168,176)(169,175)(170,174)(171,173)(177,211)
(178,210)(179,220)(180,219)(181,218)(182,217)(183,216)(184,215)(185,214)
(186,213)(187,212)(188,200)(189,199)(190,209)(191,208)(192,207)(193,206)
(194,205)(195,204)(196,203)(197,202)(198,201);
s3 := Sym(220)!( 1, 12)( 2, 13)( 3, 14)( 4, 15)( 5, 16)( 6, 17)( 7, 18)
( 8, 19)( 9, 20)( 10, 21)( 11, 22)( 23, 45)( 24, 46)( 25, 47)( 26, 48)
( 27, 49)( 28, 50)( 29, 51)( 30, 52)( 31, 53)( 32, 54)( 33, 55)( 56, 67)
( 57, 68)( 58, 69)( 59, 70)( 60, 71)( 61, 72)( 62, 73)( 63, 74)( 64, 75)
( 65, 76)( 66, 77)( 78,100)( 79,101)( 80,102)( 81,103)( 82,104)( 83,105)
( 84,106)( 85,107)( 86,108)( 87,109)( 88,110)(111,122)(112,123)(113,124)
(114,125)(115,126)(116,127)(117,128)(118,129)(119,130)(120,131)(121,132)
(133,155)(134,156)(135,157)(136,158)(137,159)(138,160)(139,161)(140,162)
(141,163)(142,164)(143,165)(166,177)(167,178)(168,179)(169,180)(170,181)
(171,182)(172,183)(173,184)(174,185)(175,186)(176,187)(188,210)(189,211)
(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218)(197,219)
(198,220);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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