Polytope of Type {6,18}

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

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

Permutation Representation (Magma) :

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

References : None.
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