Polytope of Type {2,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,6,4}*288
if this polytope has a name.
Group : SmallGroup(288,1031)
Rank : 4
Schlafli Type : {2,6,4}
Number of vertices, edges, etc : 2, 18, 36, 12
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,6,4,2} of size 576
   {2,6,4,4} of size 1152
   {2,6,4,6} of size 1728
Vertex Figure Of :
   {2,2,6,4} of size 576
   {3,2,6,4} of size 864
   {4,2,6,4} of size 1152
   {5,2,6,4} of size 1440
   {6,2,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,6,4}*144
   9-fold quotients : {2,2,4}*32
   18-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,6,8}*576, {4,6,4}*576b, {2,12,4}*576
   3-fold covers : {2,6,4}*864a, {2,6,12}*864e, {2,6,12}*864f, {6,6,4}*864j, {2,6,4}*864b, {2,6,12}*864h, {2,6,12}*864i
   4-fold covers : {4,12,4}*1152a, {2,24,4}*1152a, {2,12,8}*1152a, {2,24,4}*1152b, {2,12,8}*1152b, {2,12,4}*1152, {8,6,4}*1152a, {4,6,8}*1152b, {2,6,16}*1152
   5-fold covers : {10,6,4}*1440, {2,30,4}*1440, {2,6,20}*1440
   6-fold covers : {2,6,8}*1728a, {2,6,24}*1728d, {2,6,24}*1728e, {4,6,4}*1728a, {4,6,12}*1728f, {4,6,12}*1728g, {2,12,4}*1728b, {2,12,12}*1728f, {2,12,12}*1728g, {6,6,8}*1728f, {12,6,4}*1728k, {2,12,4}*1728d, {2,12,12}*1728j, {2,6,8}*1728b, {2,6,24}*1728g, {6,12,4}*1728n, {4,6,4}*1728d, {4,6,12}*1728m, {2,6,24}*1728h, {4,6,12}*1728n, {2,12,12}*1728l
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)(12,21)
(13,23)(14,22)(15,27)(16,29)(17,28)(18,24)(19,26)(20,25)(39,66)(40,68)(41,67)
(42,72)(43,74)(44,73)(45,69)(46,71)(47,70)(48,57)(49,59)(50,58)(51,63)(52,65)
(53,64)(54,60)(55,62)(56,61);;
s2 := ( 3, 6)( 4, 7)( 5, 8)(12,15)(13,16)(14,17)(21,24)(22,25)(23,26)(30,33)
(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)(44,50)(45,54)(46,55)(47,56)
(57,69)(58,70)(59,71)(60,66)(61,67)(62,68)(63,72)(64,73)(65,74);;
s3 := ( 3,39)( 4,42)( 5,45)( 6,40)( 7,43)( 8,46)( 9,41)(10,44)(11,47)(12,48)
(13,51)(14,54)(15,49)(16,52)(17,55)(18,50)(19,53)(20,56)(21,57)(22,60)(23,63)
(24,58)(25,61)(26,64)(27,59)(28,62)(29,65)(30,66)(31,69)(32,72)(33,67)(34,70)
(35,73)(36,68)(37,71)(38,74);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(74)!(1,2);
s1 := Sym(74)!( 3,30)( 4,32)( 5,31)( 6,36)( 7,38)( 8,37)( 9,33)(10,35)(11,34)
(12,21)(13,23)(14,22)(15,27)(16,29)(17,28)(18,24)(19,26)(20,25)(39,66)(40,68)
(41,67)(42,72)(43,74)(44,73)(45,69)(46,71)(47,70)(48,57)(49,59)(50,58)(51,63)
(52,65)(53,64)(54,60)(55,62)(56,61);
s2 := Sym(74)!( 3, 6)( 4, 7)( 5, 8)(12,15)(13,16)(14,17)(21,24)(22,25)(23,26)
(30,33)(31,34)(32,35)(39,51)(40,52)(41,53)(42,48)(43,49)(44,50)(45,54)(46,55)
(47,56)(57,69)(58,70)(59,71)(60,66)(61,67)(62,68)(63,72)(64,73)(65,74);
s3 := Sym(74)!( 3,39)( 4,42)( 5,45)( 6,40)( 7,43)( 8,46)( 9,41)(10,44)(11,47)
(12,48)(13,51)(14,54)(15,49)(16,52)(17,55)(18,50)(19,53)(20,56)(21,57)(22,60)
(23,63)(24,58)(25,61)(26,64)(27,59)(28,62)(29,65)(30,66)(31,69)(32,72)(33,67)
(34,70)(35,73)(36,68)(37,71)(38,74);
poly := sub<Sym(74)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope