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