Polytope of Type {216}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {216}*432
Also Known As : 216-gon, {216}. if this polytope has another name.
Group : SmallGroup(432,8)
Rank : 2
Schlafli Type : {216}
Number of vertices, edges, etc : 216, 216
Order of s0s1 : 216
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{216,2} of size 864
{216,4} of size 1728
{216,4} of size 1728
{216,4} of size 1728
{216,4} of size 1728
Vertex Figure Of :
{2,216} of size 864
{4,216} of size 1728
{4,216} of size 1728
{4,216} of size 1728
{4,216} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {108}*216
3-fold quotients : {72}*144
4-fold quotients : {54}*108
6-fold quotients : {36}*72
8-fold quotients : {27}*54
9-fold quotients : {24}*48
12-fold quotients : {18}*36
18-fold quotients : {12}*24
24-fold quotients : {9}*18
27-fold quotients : {8}*16
36-fold quotients : {6}*12
54-fold quotients : {4}*8
72-fold quotients : {3}*6
108-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {432}*864
3-fold covers : {648}*1296
4-fold covers : {864}*1728
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 31, 35)( 32, 34)
( 33, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 53)
( 44, 52)( 45, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)( 60, 90)
( 61, 86)( 62, 85)( 63, 87)( 64,103)( 65,105)( 66,104)( 67,100)( 68,102)
( 69,101)( 70,107)( 71,106)( 72,108)( 73, 94)( 74, 96)( 75, 95)( 76, 91)
( 77, 93)( 78, 92)( 79, 98)( 80, 97)( 81, 99)(109,163)(110,165)(111,164)
(112,170)(113,169)(114,171)(115,167)(116,166)(117,168)(118,184)(119,186)
(120,185)(121,181)(122,183)(123,182)(124,188)(125,187)(126,189)(127,175)
(128,177)(129,176)(130,172)(131,174)(132,173)(133,179)(134,178)(135,180)
(136,190)(137,192)(138,191)(139,197)(140,196)(141,198)(142,194)(143,193)
(144,195)(145,211)(146,213)(147,212)(148,208)(149,210)(150,209)(151,215)
(152,214)(153,216)(154,202)(155,204)(156,203)(157,199)(158,201)(159,200)
(160,206)(161,205)(162,207);;
s1 := ( 1,118)( 2,120)( 3,119)( 4,125)( 5,124)( 6,126)( 7,122)( 8,121)
( 9,123)( 10,109)( 11,111)( 12,110)( 13,116)( 14,115)( 15,117)( 16,113)
( 17,112)( 18,114)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)( 24,128)
( 25,134)( 26,133)( 27,135)( 28,145)( 29,147)( 30,146)( 31,152)( 32,151)
( 33,153)( 34,149)( 35,148)( 36,150)( 37,136)( 38,138)( 39,137)( 40,143)
( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,157)( 47,159)( 48,158)
( 49,154)( 50,156)( 51,155)( 52,161)( 53,160)( 54,162)( 55,199)( 56,201)
( 57,200)( 58,206)( 59,205)( 60,207)( 61,203)( 62,202)( 63,204)( 64,190)
( 65,192)( 66,191)( 67,197)( 68,196)( 69,198)( 70,194)( 71,193)( 72,195)
( 73,211)( 74,213)( 75,212)( 76,208)( 77,210)( 78,209)( 79,215)( 80,214)
( 81,216)( 82,172)( 83,174)( 84,173)( 85,179)( 86,178)( 87,180)( 88,176)
( 89,175)( 90,177)( 91,163)( 92,165)( 93,164)( 94,170)( 95,169)( 96,171)
( 97,167)( 98,166)( 99,168)(100,184)(101,186)(102,185)(103,181)(104,183)
(105,182)(106,188)(107,187)(108,189);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)( 10, 22)( 11, 24)( 12, 23)
( 13, 19)( 14, 21)( 15, 20)( 16, 26)( 17, 25)( 18, 27)( 29, 30)( 31, 35)
( 32, 34)( 33, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)
( 43, 53)( 44, 52)( 45, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 89)( 59, 88)
( 60, 90)( 61, 86)( 62, 85)( 63, 87)( 64,103)( 65,105)( 66,104)( 67,100)
( 68,102)( 69,101)( 70,107)( 71,106)( 72,108)( 73, 94)( 74, 96)( 75, 95)
( 76, 91)( 77, 93)( 78, 92)( 79, 98)( 80, 97)( 81, 99)(109,163)(110,165)
(111,164)(112,170)(113,169)(114,171)(115,167)(116,166)(117,168)(118,184)
(119,186)(120,185)(121,181)(122,183)(123,182)(124,188)(125,187)(126,189)
(127,175)(128,177)(129,176)(130,172)(131,174)(132,173)(133,179)(134,178)
(135,180)(136,190)(137,192)(138,191)(139,197)(140,196)(141,198)(142,194)
(143,193)(144,195)(145,211)(146,213)(147,212)(148,208)(149,210)(150,209)
(151,215)(152,214)(153,216)(154,202)(155,204)(156,203)(157,199)(158,201)
(159,200)(160,206)(161,205)(162,207);
s1 := Sym(216)!( 1,118)( 2,120)( 3,119)( 4,125)( 5,124)( 6,126)( 7,122)
( 8,121)( 9,123)( 10,109)( 11,111)( 12,110)( 13,116)( 14,115)( 15,117)
( 16,113)( 17,112)( 18,114)( 19,130)( 20,132)( 21,131)( 22,127)( 23,129)
( 24,128)( 25,134)( 26,133)( 27,135)( 28,145)( 29,147)( 30,146)( 31,152)
( 32,151)( 33,153)( 34,149)( 35,148)( 36,150)( 37,136)( 38,138)( 39,137)
( 40,143)( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,157)( 47,159)
( 48,158)( 49,154)( 50,156)( 51,155)( 52,161)( 53,160)( 54,162)( 55,199)
( 56,201)( 57,200)( 58,206)( 59,205)( 60,207)( 61,203)( 62,202)( 63,204)
( 64,190)( 65,192)( 66,191)( 67,197)( 68,196)( 69,198)( 70,194)( 71,193)
( 72,195)( 73,211)( 74,213)( 75,212)( 76,208)( 77,210)( 78,209)( 79,215)
( 80,214)( 81,216)( 82,172)( 83,174)( 84,173)( 85,179)( 86,178)( 87,180)
( 88,176)( 89,175)( 90,177)( 91,163)( 92,165)( 93,164)( 94,170)( 95,169)
( 96,171)( 97,167)( 98,166)( 99,168)(100,184)(101,186)(102,185)(103,181)
(104,183)(105,182)(106,188)(107,187)(108,189);
poly := sub<Sym(216)|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 >;
References : None.
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