Part of the Atlas of Small Regular Polytopes

Polytope of Type {216}

Atlas Canonical Name {216}*432

Overview

Group
SmallGroup(432,8)
Rank
2
Schläfli Type
{216}
Vertices, edges, …
216, 216
Order of s0s1
216
Also known as
216-gon, {216}. if this polytope has another name.

Special Properties

  • Universal
  • Spherical
  • Locally Spherical
  • Orientable
  • Self-Dual

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

18-fold

24-fold

27-fold

36-fold

54-fold

72-fold

108-fold

Covers minimal covers in bold

2-fold

3-fold

4-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

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.

to this polytope.