Polytope of Type {68}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {68}*136
Also Known As : 68-gon, {68}. if this polytope has another name.
Group : SmallGroup(136,6)
Rank : 2
Schlafli Type : {68}
Number of vertices, edges, etc : 68, 68
Order of s0s1 : 68
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{68,2} of size 272
{68,4} of size 544
{68,6} of size 816
{68,6} of size 816
{68,8} of size 1088
{68,8} of size 1088
{68,4} of size 1088
{68,6} of size 1224
{68,10} of size 1360
{68,12} of size 1632
{68,6} of size 1632
{68,14} of size 1904
Vertex Figure Of :
{2,68} of size 272
{4,68} of size 544
{6,68} of size 816
{6,68} of size 816
{8,68} of size 1088
{8,68} of size 1088
{4,68} of size 1088
{6,68} of size 1224
{10,68} of size 1360
{12,68} of size 1632
{6,68} of size 1632
{14,68} of size 1904
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {34}*68
4-fold quotients : {17}*34
17-fold quotients : {4}*8
34-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {136}*272
3-fold covers : {204}*408
4-fold covers : {272}*544
5-fold covers : {340}*680
6-fold covers : {408}*816
7-fold covers : {476}*952
8-fold covers : {544}*1088
9-fold covers : {612}*1224
10-fold covers : {680}*1360
11-fold covers : {748}*1496
12-fold covers : {816}*1632
13-fold covers : {884}*1768
14-fold covers : {952}*1904
Permutation Representation (GAP) :
s0 := ( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)(20,33)
(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,52)(36,68)(37,67)(38,66)(39,65)
(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)(50,54)
(51,53);;
s1 := ( 1,36)( 2,35)( 3,51)( 4,50)( 5,49)( 6,48)( 7,47)( 8,46)( 9,45)(10,44)
(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,53)(19,52)(20,68)(21,67)
(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)(32,56)
(33,55)(34,54);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(68)!( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,34)
(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(26,27)(35,52)(36,68)(37,67)(38,66)
(39,65)(40,64)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,56)(49,55)
(50,54)(51,53);
s1 := Sym(68)!( 1,36)( 2,35)( 3,51)( 4,50)( 5,49)( 6,48)( 7,47)( 8,46)( 9,45)
(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,53)(19,52)(20,68)
(21,67)(22,66)(23,65)(24,64)(25,63)(26,62)(27,61)(28,60)(29,59)(30,58)(31,57)
(32,56)(33,55)(34,54);
poly := sub<Sym(68)|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 >;
References : None.
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