Polytope of Type {228}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {228}*456
Also Known As : 228-gon, {228}. if this polytope has another name.
Group : SmallGroup(456,36)
Rank : 2
Schlafli Type : {228}
Number of vertices, edges, etc : 228, 228
Order of s0s1 : 228
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{228,2} of size 912
{228,4} of size 1824
{228,4} of size 1824
{228,4} of size 1824
Vertex Figure Of :
{2,228} of size 912
{4,228} of size 1824
{4,228} of size 1824
{4,228} of size 1824
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {114}*228
3-fold quotients : {76}*152
4-fold quotients : {57}*114
6-fold quotients : {38}*76
12-fold quotients : {19}*38
19-fold quotients : {12}*24
38-fold quotients : {6}*12
57-fold quotients : {4}*8
76-fold quotients : {3}*6
114-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {456}*912
3-fold covers : {684}*1368
4-fold covers : {912}*1824
Irregular Quotients (of which this is a minimal cover):
None.
Permutation Representation (GAP) :
s0 := ( 2, 19)( 3, 18)( 4, 17)( 5, 16)( 6, 15)( 7, 14)( 8, 13)( 9, 12)( 10, 11)( 20, 39)( 21, 57)( 22, 56)( 23, 55)( 24, 54)( 25, 53)( 26, 52)( 27, 51)( 28, 50)( 29, 49)( 30, 48)( 31, 47)( 32, 46)( 33, 45)( 34, 44)( 35, 43)( 36, 42)( 37, 41)( 38, 40)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 67, 68)( 77, 96)( 78,114)( 79,113)( 80,112)( 81,111)( 82,110)( 83,109)( 84,108)( 85,107)( 86,106)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,100)( 93, 99)( 94, 98)( 95, 97)(115,172)(116,190)(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)(132,174)(133,173)(134,210)(135,228)(136,227)(137,226)(138,225)(139,224)(140,223)(141,222)(142,221)(143,220)(144,219)(145,218)(146,217)(147,216)(148,215)(149,214)(150,213)(151,212)(152,211)(153,191)(154,209)(155,208)(156,207)(157,206)(158,205)(159,204)(160,203)(161,202)(162,201)(163,200)(164,199)(165,198)(166,197)(167,196)(168,195)(169,194)(170,193)(171,192);;
s1 := ( 1,135)( 2,134)( 3,152)( 4,151)( 5,150)( 6,149)( 7,148)( 8,147)( 9,146)( 10,145)( 11,144)( 12,143)( 13,142)( 14,141)( 15,140)( 16,139)( 17,138)( 18,137)( 19,136)( 20,116)( 21,115)( 22,133)( 23,132)( 24,131)( 25,130)( 26,129)( 27,128)( 28,127)( 29,126)( 30,125)( 31,124)( 32,123)( 33,122)( 34,121)( 35,120)( 36,119)( 37,118)( 38,117)( 39,154)( 40,153)( 41,171)( 42,170)( 43,169)( 44,168)( 45,167)( 46,166)( 47,165)( 48,164)( 49,163)( 50,162)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)( 57,155)( 58,192)( 59,191)( 60,209)( 61,208)( 62,207)( 63,206)( 64,205)( 65,204)( 66,203)( 67,202)( 68,201)( 69,200)( 70,199)( 71,198)( 72,197)( 73,196)( 74,195)( 75,194)( 76,193)( 77,173)( 78,172)( 79,190)( 80,189)( 81,188)( 82,187)( 83,186)( 84,185)( 85,184)( 86,183)( 87,182)( 88,181)( 89,180)( 90,179)( 91,178)( 92,177)( 93,176)( 94,175)( 95,174)( 96,211)( 97,210)( 98,228)( 99,227)(100,226)(101,225)(102,224)(103,223)(104,222)(105,221)(106,220)(107,219)(108,218)(109,217)(110,216)(111,215)(112,214)(113,213)(114,212);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(228)!( 2, 19)( 3, 18)( 4, 17)( 5, 16)( 6, 15)( 7, 14)( 8, 13)( 9, 12)( 10, 11)( 20, 39)( 21, 57)( 22, 56)( 23, 55)( 24, 54)( 25, 53)( 26, 52)( 27, 51)( 28, 50)( 29, 49)( 30, 48)( 31, 47)( 32, 46)( 33, 45)( 34, 44)( 35, 43)( 36, 42)( 37, 41)( 38, 40)( 59, 76)( 60, 75)( 61, 74)( 62, 73)( 63, 72)( 64, 71)( 65, 70)( 66, 69)( 67, 68)( 77, 96)( 78,114)( 79,113)( 80,112)( 81,111)( 82,110)( 83,109)( 84,108)( 85,107)( 86,106)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)( 92,100)( 93, 99)( 94, 98)( 95, 97)(115,172)(116,190)(117,189)(118,188)(119,187)(120,186)(121,185)(122,184)(123,183)(124,182)(125,181)(126,180)(127,179)(128,178)(129,177)(130,176)(131,175)(132,174)(133,173)(134,210)(135,228)(136,227)(137,226)(138,225)(139,224)(140,223)(141,222)(142,221)(143,220)(144,219)(145,218)(146,217)(147,216)(148,215)(149,214)(150,213)(151,212)(152,211)(153,191)(154,209)(155,208)(156,207)(157,206)(158,205)(159,204)(160,203)(161,202)(162,201)(163,200)(164,199)(165,198)(166,197)(167,196)(168,195)(169,194)(170,193)(171,192);
s1 := Sym(228)!( 1,135)( 2,134)( 3,152)( 4,151)( 5,150)( 6,149)( 7,148)( 8,147)( 9,146)( 10,145)( 11,144)( 12,143)( 13,142)( 14,141)( 15,140)( 16,139)( 17,138)( 18,137)( 19,136)( 20,116)( 21,115)( 22,133)( 23,132)( 24,131)( 25,130)( 26,129)( 27,128)( 28,127)( 29,126)( 30,125)( 31,124)( 32,123)( 33,122)( 34,121)( 35,120)( 36,119)( 37,118)( 38,117)( 39,154)( 40,153)( 41,171)( 42,170)( 43,169)( 44,168)( 45,167)( 46,166)( 47,165)( 48,164)( 49,163)( 50,162)( 51,161)( 52,160)( 53,159)( 54,158)( 55,157)( 56,156)( 57,155)( 58,192)( 59,191)( 60,209)( 61,208)( 62,207)( 63,206)( 64,205)( 65,204)( 66,203)( 67,202)( 68,201)( 69,200)( 70,199)( 71,198)( 72,197)( 73,196)( 74,195)( 75,194)( 76,193)( 77,173)( 78,172)( 79,190)( 80,189)( 81,188)( 82,187)( 83,186)( 84,185)( 85,184)( 86,183)( 87,182)( 88,181)( 89,180)( 90,179)( 91,178)( 92,177)( 93,176)( 94,175)( 95,174)( 96,211)( 97,210)( 98,228)( 99,227)(100,226)(101,225)(102,224)(103,223)(104,222)(105,221)(106,220)(107,219)(108,218)(109,217)(110,216)(111,215)(112,214)(113,213)(114,212);
poly := sub<Sym(228)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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