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Polytope of Type {2,8,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,18}*576
if this polytope has a name.
Group : SmallGroup(576,1738)
Rank : 4
Schlafli Type : {2,8,18}
Number of vertices, edges, etc : 2, 8, 72, 18
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,8,18,2} of size 1152
Vertex Figure Of :
   {2,2,8,18} of size 1152
   {3,2,8,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,18}*288a
   3-fold quotients : {2,8,6}*192
   4-fold quotients : {2,2,18}*144
   6-fold quotients : {2,4,6}*96a
   8-fold quotients : {2,2,9}*72
   9-fold quotients : {2,8,2}*64
   12-fold quotients : {2,2,6}*48
   18-fold quotients : {2,4,2}*32
   24-fold quotients : {2,2,3}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,8,18}*1152a, {2,8,36}*1152a, {2,16,18}*1152
   3-fold covers : {2,8,54}*1728, {2,24,18}*1728a, {6,8,18}*1728, {2,24,18}*1728b
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)(39,57)
(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)
(51,69)(52,70)(53,71)(54,72)(55,73)(56,74);;
s2 := ( 3,39)( 4,41)( 5,40)( 6,46)( 7,45)( 8,47)( 9,43)(10,42)(11,44)(12,48)
(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,66)(22,68)(23,67)
(24,73)(25,72)(26,74)(27,70)(28,69)(29,71)(30,57)(31,59)(32,58)(33,64)(34,63)
(35,65)(36,61)(37,60)(38,62);;
s3 := ( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,24)(22,26)
(23,25)(27,28)(30,33)(31,35)(32,34)(36,37)(39,42)(40,44)(41,43)(45,46)(48,51)
(49,53)(50,52)(54,55)(57,60)(58,62)(59,61)(63,64)(66,69)(67,71)(68,70)
(72,73);;
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(74)!(1,2);
s1 := Sym(74)!(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36)(28,37)(29,38)
(39,57)(40,58)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)
(50,68)(51,69)(52,70)(53,71)(54,72)(55,73)(56,74);
s2 := Sym(74)!( 3,39)( 4,41)( 5,40)( 6,46)( 7,45)( 8,47)( 9,43)(10,42)(11,44)
(12,48)(13,50)(14,49)(15,55)(16,54)(17,56)(18,52)(19,51)(20,53)(21,66)(22,68)
(23,67)(24,73)(25,72)(26,74)(27,70)(28,69)(29,71)(30,57)(31,59)(32,58)(33,64)
(34,63)(35,65)(36,61)(37,60)(38,62);
s3 := Sym(74)!( 3, 6)( 4, 8)( 5, 7)( 9,10)(12,15)(13,17)(14,16)(18,19)(21,24)
(22,26)(23,25)(27,28)(30,33)(31,35)(32,34)(36,37)(39,42)(40,44)(41,43)(45,46)
(48,51)(49,53)(50,52)(54,55)(57,60)(58,62)(59,61)(63,64)(66,69)(67,71)(68,70)
(72,73);
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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