Polytope of Type {108}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {108}*216
Also Known As : 108-gon, {108}. if this polytope has another name.
Group : SmallGroup(216,6)
Rank : 2
Schlafli Type : {108}
Number of vertices, edges, etc : 108, 108
Order of s0s1 : 108
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{108,2} of size 432
{108,4} of size 864
{108,4} of size 864
{108,4} of size 864
{108,6} of size 1296
{108,6} of size 1296
{108,6} of size 1296
{108,4} of size 1728
{108,8} of size 1728
{108,8} of size 1728
{108,4} of size 1728
{108,4} of size 1728
{108,6} of size 1944
{108,6} of size 1944
{108,6} of size 1944
Vertex Figure Of :
{2,108} of size 432
{4,108} of size 864
{4,108} of size 864
{4,108} of size 864
{6,108} of size 1296
{6,108} of size 1296
{6,108} of size 1296
{4,108} of size 1728
{8,108} of size 1728
{8,108} of size 1728
{4,108} of size 1728
{4,108} of size 1728
{6,108} of size 1944
{6,108} of size 1944
{6,108} of size 1944
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {54}*108
3-fold quotients : {36}*72
4-fold quotients : {27}*54
6-fold quotients : {18}*36
9-fold quotients : {12}*24
12-fold quotients : {9}*18
18-fold quotients : {6}*12
27-fold quotients : {4}*8
36-fold quotients : {3}*6
54-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {216}*432
3-fold covers : {324}*648
4-fold covers : {432}*864
5-fold covers : {540}*1080
6-fold covers : {648}*1296
7-fold covers : {756}*1512
8-fold covers : {864}*1728
9-fold covers : {972}*1944
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);;
s1 := ( 1, 64)( 2, 66)( 3, 65)( 4, 71)( 5, 70)( 6, 72)( 7, 68)( 8, 67)
( 9, 69)( 10, 55)( 11, 57)( 12, 56)( 13, 62)( 14, 61)( 15, 63)( 16, 59)
( 17, 58)( 18, 60)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)( 24, 74)
( 25, 80)( 26, 79)( 27, 81)( 28, 91)( 29, 93)( 30, 92)( 31, 98)( 32, 97)
( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)( 40, 89)
( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,103)( 47,105)( 48,104)
( 49,100)( 50,102)( 51,101)( 52,107)( 53,106)( 54,108);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(108)!( 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);
s1 := Sym(108)!( 1, 64)( 2, 66)( 3, 65)( 4, 71)( 5, 70)( 6, 72)( 7, 68)
( 8, 67)( 9, 69)( 10, 55)( 11, 57)( 12, 56)( 13, 62)( 14, 61)( 15, 63)
( 16, 59)( 17, 58)( 18, 60)( 19, 76)( 20, 78)( 21, 77)( 22, 73)( 23, 75)
( 24, 74)( 25, 80)( 26, 79)( 27, 81)( 28, 91)( 29, 93)( 30, 92)( 31, 98)
( 32, 97)( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37, 82)( 38, 84)( 39, 83)
( 40, 89)( 41, 88)( 42, 90)( 43, 86)( 44, 85)( 45, 87)( 46,103)( 47,105)
( 48,104)( 49,100)( 50,102)( 51,101)( 52,107)( 53,106)( 54,108);
poly := sub<Sym(108)|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 >;
References : None.
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