Polytope of Type {6,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,6}*432g
if this polytope has a name.
Group : SmallGroup(432,759)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 18, 18, 6
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,6,2} of size 864
   {6,6,6,3} of size 1296
   {6,6,6,4} of size 1728
   {6,6,6,3} of size 1728
   {6,6,6,4} of size 1728
Vertex Figure Of :
   {2,6,6,6} of size 864
   {4,6,6,6} of size 1728
   {4,6,6,6} of size 1728
   {4,6,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,6}*216b
   3-fold quotients : {6,2,6}*144, {6,6,2}*144c
   6-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
   9-fold quotients : {2,2,6}*48, {6,2,2}*48
   12-fold quotients : {3,2,3}*36
   18-fold quotients : {2,2,3}*24, {3,2,2}*24
   27-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,6,6}*864d, {6,6,12}*864e, {6,12,6}*864f
   3-fold covers : {6,6,18}*1296e, {18,6,6}*1296e, {6,6,6}*1296c, {6,6,6}*1296o, {6,6,6}*1296p, {6,6,6}*1296s
   4-fold covers : {24,6,6}*1728d, {6,6,24}*1728e, {12,6,12}*1728f, {12,12,6}*1728f, {6,24,6}*1728g, {6,12,12}*1728g, {6,6,6}*1728f, {6,12,6}*1728h, {6,12,6}*1728l
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)(22,25)
(23,27)(24,26)(29,30)(31,34)(32,36)(33,35)(38,39)(40,43)(41,45)(42,44)(47,48)
(49,52)(50,54)(51,53);;
s1 := ( 1,32)( 2,31)( 3,33)( 4,29)( 5,28)( 6,30)( 7,35)( 8,34)( 9,36)(10,41)
(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,50)(20,49)(21,51)
(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)
(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)(42,50)
(43,52)(44,54)(45,53);;
s3 := ( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(28,37)
(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(54)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)(11,12)(13,16)(14,18)(15,17)(20,21)
(22,25)(23,27)(24,26)(29,30)(31,34)(32,36)(33,35)(38,39)(40,43)(41,45)(42,44)
(47,48)(49,52)(50,54)(51,53);
s1 := Sym(54)!( 1,32)( 2,31)( 3,33)( 4,29)( 5,28)( 6,30)( 7,35)( 8,34)( 9,36)
(10,41)(11,40)(12,42)(13,38)(14,37)(15,39)(16,44)(17,43)(18,45)(19,50)(20,49)
(21,51)(22,47)(23,46)(24,48)(25,53)(26,52)(27,54);
s2 := Sym(54)!( 2, 3)( 5, 6)( 8, 9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)
(16,25)(17,27)(18,26)(29,30)(32,33)(35,36)(37,46)(38,48)(39,47)(40,49)(41,51)
(42,50)(43,52)(44,54)(45,53);
s3 := Sym(54)!( 1,10)( 2,11)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)
(28,37)(29,38)(30,39)(31,40)(32,41)(33,42)(34,43)(35,44)(36,45);
poly := sub<Sym(54)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope