Polytope of Type {84}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84}*168
Also Known As : 84-gon, {84}. if this polytope has another name.
Group : SmallGroup(168,36)
Rank : 2
Schlafli Type : {84}
Number of vertices, edges, etc : 84, 84
Order of s0s1 : 84
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{84,2} of size 336
{84,4} of size 672
{84,4} of size 672
{84,4} of size 672
{84,6} of size 1008
{84,6} of size 1008
{84,6} of size 1008
{84,6} of size 1008
{84,4} of size 1344
{84,8} of size 1344
{84,8} of size 1344
{84,6} of size 1344
{84,6} of size 1344
{84,4} of size 1344
{84,4} of size 1344
{84,6} of size 1512
{84,6} of size 1512
{84,6} of size 1512
{84,10} of size 1680
Vertex Figure Of :
{2,84} of size 336
{4,84} of size 672
{4,84} of size 672
{4,84} of size 672
{6,84} of size 1008
{6,84} of size 1008
{6,84} of size 1008
{6,84} of size 1008
{4,84} of size 1344
{8,84} of size 1344
{8,84} of size 1344
{6,84} of size 1344
{6,84} of size 1344
{4,84} of size 1344
{4,84} of size 1344
{6,84} of size 1512
{6,84} of size 1512
{6,84} of size 1512
{10,84} of size 1680
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {42}*84
3-fold quotients : {28}*56
4-fold quotients : {21}*42
6-fold quotients : {14}*28
7-fold quotients : {12}*24
12-fold quotients : {7}*14
14-fold quotients : {6}*12
21-fold quotients : {4}*8
28-fold quotients : {3}*6
42-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {168}*336
3-fold covers : {252}*504
4-fold covers : {336}*672
5-fold covers : {420}*840
6-fold covers : {504}*1008
7-fold covers : {588}*1176
8-fold covers : {672}*1344
9-fold covers : {756}*1512
10-fold covers : {840}*1680
11-fold covers : {924}*1848
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)
(23,28)(24,27)(25,26)(29,36)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,64)
(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,78)(51,84)(52,83)(53,82)(54,81)
(55,80)(56,79)(57,71)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72);;
s1 := ( 1,51)( 2,50)( 3,56)( 4,55)( 5,54)( 6,53)( 7,52)( 8,44)( 9,43)(10,49)
(11,48)(12,47)(13,46)(14,45)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)(21,59)
(22,72)(23,71)(24,77)(25,76)(26,75)(27,74)(28,73)(29,65)(30,64)(31,70)(32,69)
(33,68)(34,67)(35,66)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);;
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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(84)!( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)
(14,16)(23,28)(24,27)(25,26)(29,36)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)
(43,64)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,78)(51,84)(52,83)(53,82)
(54,81)(55,80)(56,79)(57,71)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72);
s1 := Sym(84)!( 1,51)( 2,50)( 3,56)( 4,55)( 5,54)( 6,53)( 7,52)( 8,44)( 9,43)
(10,49)(11,48)(12,47)(13,46)(14,45)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)
(21,59)(22,72)(23,71)(24,77)(25,76)(26,75)(27,74)(28,73)(29,65)(30,64)(31,70)
(32,69)(33,68)(34,67)(35,66)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);
poly := sub<Sym(84)|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 >;
References : None.
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