Polytope of Type {222}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {222}*444
Also Known As : 222-gon, {222}. if this polytope has another name.
Group : SmallGroup(444,17)
Rank : 2
Schlafli Type : {222}
Number of vertices, edges, etc : 222, 222
Order of s0s1 : 222
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{222,2} of size 888
{222,4} of size 1776
{222,4} of size 1776
{222,4} of size 1776
Vertex Figure Of :
{2,222} of size 888
{4,222} of size 1776
{4,222} of size 1776
{4,222} of size 1776
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {111}*222
3-fold quotients : {74}*148
6-fold quotients : {37}*74
37-fold quotients : {6}*12
74-fold quotients : {3}*6
111-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {444}*888
3-fold covers : {666}*1332
4-fold covers : {888}*1776
Permutation Representation (GAP) :
s0 := ( 2, 37)( 3, 36)( 4, 35)( 5, 34)( 6, 33)( 7, 32)( 8, 31)( 9, 30)
( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)( 17, 22)
( 18, 21)( 19, 20)( 38, 75)( 39,111)( 40,110)( 41,109)( 42,108)( 43,107)
( 44,106)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)( 51, 99)
( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 93)( 58, 92)( 59, 91)
( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)( 67, 83)
( 68, 82)( 69, 81)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)(113,148)
(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)
(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)
(130,131)(149,186)(150,222)(151,221)(152,220)(153,219)(154,218)(155,217)
(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)(163,209)
(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)(171,201)
(172,200)(173,199)(174,198)(175,197)(176,196)(177,195)(178,194)(179,193)
(180,192)(181,191)(182,190)(183,189)(184,188)(185,187);;
s1 := ( 1,150)( 2,149)( 3,185)( 4,184)( 5,183)( 6,182)( 7,181)( 8,180)
( 9,179)( 10,178)( 11,177)( 12,176)( 13,175)( 14,174)( 15,173)( 16,172)
( 17,171)( 18,170)( 19,169)( 20,168)( 21,167)( 22,166)( 23,165)( 24,164)
( 25,163)( 26,162)( 27,161)( 28,160)( 29,159)( 30,158)( 31,157)( 32,156)
( 33,155)( 34,154)( 35,153)( 36,152)( 37,151)( 38,113)( 39,112)( 40,148)
( 41,147)( 42,146)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,140)
( 49,139)( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,133)( 56,132)
( 57,131)( 58,130)( 59,129)( 60,128)( 61,127)( 62,126)( 63,125)( 64,124)
( 65,123)( 66,122)( 67,121)( 68,120)( 69,119)( 70,118)( 71,117)( 72,116)
( 73,115)( 74,114)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)( 80,219)
( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)( 88,211)
( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)( 96,203)
( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)(104,195)
(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(222)!( 2, 37)( 3, 36)( 4, 35)( 5, 34)( 6, 33)( 7, 32)( 8, 31)
( 9, 30)( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)
( 17, 22)( 18, 21)( 19, 20)( 38, 75)( 39,111)( 40,110)( 41,109)( 42,108)
( 43,107)( 44,106)( 45,105)( 46,104)( 47,103)( 48,102)( 49,101)( 50,100)
( 51, 99)( 52, 98)( 53, 97)( 54, 96)( 55, 95)( 56, 94)( 57, 93)( 58, 92)
( 59, 91)( 60, 90)( 61, 89)( 62, 88)( 63, 87)( 64, 86)( 65, 85)( 66, 84)
( 67, 83)( 68, 82)( 69, 81)( 70, 80)( 71, 79)( 72, 78)( 73, 77)( 74, 76)
(113,148)(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)
(121,140)(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)
(129,132)(130,131)(149,186)(150,222)(151,221)(152,220)(153,219)(154,218)
(155,217)(156,216)(157,215)(158,214)(159,213)(160,212)(161,211)(162,210)
(163,209)(164,208)(165,207)(166,206)(167,205)(168,204)(169,203)(170,202)
(171,201)(172,200)(173,199)(174,198)(175,197)(176,196)(177,195)(178,194)
(179,193)(180,192)(181,191)(182,190)(183,189)(184,188)(185,187);
s1 := Sym(222)!( 1,150)( 2,149)( 3,185)( 4,184)( 5,183)( 6,182)( 7,181)
( 8,180)( 9,179)( 10,178)( 11,177)( 12,176)( 13,175)( 14,174)( 15,173)
( 16,172)( 17,171)( 18,170)( 19,169)( 20,168)( 21,167)( 22,166)( 23,165)
( 24,164)( 25,163)( 26,162)( 27,161)( 28,160)( 29,159)( 30,158)( 31,157)
( 32,156)( 33,155)( 34,154)( 35,153)( 36,152)( 37,151)( 38,113)( 39,112)
( 40,148)( 41,147)( 42,146)( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)
( 48,140)( 49,139)( 50,138)( 51,137)( 52,136)( 53,135)( 54,134)( 55,133)
( 56,132)( 57,131)( 58,130)( 59,129)( 60,128)( 61,127)( 62,126)( 63,125)
( 64,124)( 65,123)( 66,122)( 67,121)( 68,120)( 69,119)( 70,118)( 71,117)
( 72,116)( 73,115)( 74,114)( 75,187)( 76,186)( 77,222)( 78,221)( 79,220)
( 80,219)( 81,218)( 82,217)( 83,216)( 84,215)( 85,214)( 86,213)( 87,212)
( 88,211)( 89,210)( 90,209)( 91,208)( 92,207)( 93,206)( 94,205)( 95,204)
( 96,203)( 97,202)( 98,201)( 99,200)(100,199)(101,198)(102,197)(103,196)
(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,188);
poly := sub<Sym(222)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1*s0*s1 >;
References : None.
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