Polytope of Type {2,3,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,3,4,6}*576
if this polytope has a name.
Group : SmallGroup(576,8659)
Rank : 5
Schlafli Type : {2,3,4,6}
Number of vertices, edges, etc : 2, 6, 12, 24, 6
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,3,4,6,2} of size 1152
   {2,3,4,6,3} of size 1728
Vertex Figure Of :
   {2,2,3,4,6} of size 1152
   {3,2,3,4,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,3,4,2}*192
   4-fold quotients : {2,3,2,6}*144
   6-fold quotients : {2,3,4,2}*96
   8-fold quotients : {2,3,2,3}*72
   12-fold quotients : {2,3,2,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,3,4,12}*1152, {2,3,8,6}*1152, {2,6,4,6}*1152b
   3-fold covers : {2,3,4,18}*1728, {2,9,4,6}*1728, {2,3,12,6}*1728a, {6,3,4,6}*1728, {2,3,12,6}*1728b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 5)( 8, 9)(12,13)(15,27)(16,29)(17,28)(18,30)(19,31)(20,33)(21,32)
(22,34)(23,35)(24,37)(25,36)(26,38)(40,41)(44,45)(48,49)(51,63)(52,65)(53,64)
(54,66)(55,67)(56,69)(57,68)(58,70)(59,71)(60,73)(61,72)(62,74);;
s2 := ( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,22)(10,21)(11,23)(12,24)
(13,26)(14,25)(29,30)(33,34)(37,38)(39,51)(40,52)(41,54)(42,53)(43,55)(44,56)
(45,58)(46,57)(47,59)(48,60)(49,62)(50,61)(65,66)(69,70)(73,74);;
s3 := ( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,12)(10,11)(15,18)(16,17)(19,26)(20,25)
(21,24)(22,23)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,42)(40,41)(43,50)
(44,49)(45,48)(46,47)(51,54)(52,53)(55,62)(56,61)(57,60)(58,59)(63,66)(64,65)
(67,74)(68,73)(69,72)(70,71);;
s4 := ( 3,43)( 4,44)( 5,45)( 6,46)( 7,39)( 8,40)( 9,41)(10,42)(11,47)(12,48)
(13,49)(14,50)(15,55)(16,56)(17,57)(18,58)(19,51)(20,52)(21,53)(22,54)(23,59)
(24,60)(25,61)(26,62)(27,67)(28,68)(29,69)(30,70)(31,63)(32,64)(33,65)(34,66)
(35,71)(36,72)(37,73)(38,74);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(74)!(1,2);
s1 := Sym(74)!( 4, 5)( 8, 9)(12,13)(15,27)(16,29)(17,28)(18,30)(19,31)(20,33)
(21,32)(22,34)(23,35)(24,37)(25,36)(26,38)(40,41)(44,45)(48,49)(51,63)(52,65)
(53,64)(54,66)(55,67)(56,69)(57,68)(58,70)(59,71)(60,73)(61,72)(62,74);
s2 := Sym(74)!( 3,15)( 4,16)( 5,18)( 6,17)( 7,19)( 8,20)( 9,22)(10,21)(11,23)
(12,24)(13,26)(14,25)(29,30)(33,34)(37,38)(39,51)(40,52)(41,54)(42,53)(43,55)
(44,56)(45,58)(46,57)(47,59)(48,60)(49,62)(50,61)(65,66)(69,70)(73,74);
s3 := Sym(74)!( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,12)(10,11)(15,18)(16,17)(19,26)
(20,25)(21,24)(22,23)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,42)(40,41)
(43,50)(44,49)(45,48)(46,47)(51,54)(52,53)(55,62)(56,61)(57,60)(58,59)(63,66)
(64,65)(67,74)(68,73)(69,72)(70,71);
s4 := Sym(74)!( 3,43)( 4,44)( 5,45)( 6,46)( 7,39)( 8,40)( 9,41)(10,42)(11,47)
(12,48)(13,49)(14,50)(15,55)(16,56)(17,57)(18,58)(19,51)(20,52)(21,53)(22,54)
(23,59)(24,60)(25,61)(26,62)(27,67)(28,68)(29,69)(30,70)(31,63)(32,64)(33,65)
(34,66)(35,71)(36,72)(37,73)(38,74);
poly := sub<Sym(74)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope