Polytope of Type {18,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,6}*216a
Also Known As : {18,6|2}. if this polytope has another name.
Group : SmallGroup(216,101)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 18, 54, 6
Order of s₀s₁s₂ : 18
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,6,2} of size 432
   {18,6,3} of size 648
   {18,6,4} of size 864
   {18,6,3} of size 864
   {18,6,4} of size 864
   {18,6,6} of size 1296
   {18,6,6} of size 1296
   {18,6,6} of size 1296
   {18,6,8} of size 1728
   {18,6,4} of size 1728
   {18,6,6} of size 1728
   {18,6,9} of size 1944
   {18,6,3} of size 1944
Vertex Figure Of :
   {2,18,6} of size 432
   {4,18,6} of size 864
   {4,18,6} of size 864
   {6,18,6} of size 1296
   {6,18,6} of size 1296
   {8,18,6} of size 1728
   {4,18,6} of size 1728
   {9,18,6} of size 1944
   {3,18,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {18,2}*72, {6,6}*72a
   6-fold quotients : {9,2}*36
   9-fold quotients : {2,6}*24, {6,2}*24
   18-fold quotients : {2,3}*12, {3,2}*12
   27-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,6}*432a, {18,12}*432a
   3-fold covers : {18,18}*648a, {18,6}*648b, {54,6}*648a, {18,6}*648i
   4-fold covers : {72,6}*864a, {18,24}*864a, {36,12}*864a, {36,6}*864, {18,12}*864a
   5-fold covers : {90,6}*1080a, {18,30}*1080b
   6-fold covers : {18,36}*1296a, {36,18}*1296a, {18,12}*1296a, {36,6}*1296b, {54,12}*1296a, {108,6}*1296a, {36,6}*1296l, {18,12}*1296l
   7-fold covers : {126,6}*1512a, {18,42}*1512b
   8-fold covers : {144,6}*1728a, {18,48}*1728a, {36,12}*1728a, {72,12}*1728a, {36,24}*1728c, {72,12}*1728c, {36,24}*1728d, {36,12}*1728c, {36,6}*1728b, {72,6}*1728b, {72,6}*1728c, {36,12}*1728d, {36,12}*1728e, {18,12}*1728c, {18,24}*1728c, {18,24}*1728e, {36,12}*1728h
   9-fold covers : {18,18}*1944c, {18,54}*1944a, {54,18}*1944a, {54,6}*1944b, {18,6}*1944g, {18,18}*1944s, {18,18}*1944x, {18,6}*1944j, {54,6}*1944d, {54,6}*1944f, {162,6}*1944a, {18,18}*1944ad, {18,18}*1944af, {18,6}*1944m, {18,6}*1944n, {18,6}*1944o, {54,6}*1944g
Permutation Representation (GAP) :

s0 := ( 2, 3)( 5, 6)( 8, 9)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)(16,26)
(17,25)(18,27)(29,30)(32,33)(35,36)(37,47)(38,46)(39,48)(40,50)(41,49)(42,51)
(43,53)(44,52)(45,54);;
s1 := ( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)(19,20)
(22,26)(23,25)(24,27)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)(35,42)
(36,41)(46,47)(49,53)(50,52)(51,54);;
s2 := ( 1,31)( 2,32)( 3,33)( 4,28)( 5,29)( 6,30)( 7,34)( 8,35)( 9,36)(10,40)
(11,41)(12,42)(13,37)(14,38)(15,39)(16,43)(17,44)(18,45)(19,49)(20,50)(21,51)
(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(54)!( 2, 3)( 5, 6)( 8, 9)(10,20)(11,19)(12,21)(13,23)(14,22)(15,24)
(16,26)(17,25)(18,27)(29,30)(32,33)(35,36)(37,47)(38,46)(39,48)(40,50)(41,49)
(42,51)(43,53)(44,52)(45,54);
s1 := Sym(54)!( 1,10)( 2,12)( 3,11)( 4,16)( 5,18)( 6,17)( 7,13)( 8,15)( 9,14)
(19,20)(22,26)(23,25)(24,27)(28,37)(29,39)(30,38)(31,43)(32,45)(33,44)(34,40)
(35,42)(36,41)(46,47)(49,53)(50,52)(51,54);
s2 := Sym(54)!( 1,31)( 2,32)( 3,33)( 4,28)( 5,29)( 6,30)( 7,34)( 8,35)( 9,36)
(10,40)(11,41)(12,42)(13,37)(14,38)(15,39)(16,43)(17,44)(18,45)(19,49)(20,50)
(21,51)(22,46)(23,47)(24,48)(25,52)(26,53)(27,54);
poly := sub<Sym(54)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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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