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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,2,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47887)
Rank : 5
Schlafli Type : {4,6,2,6}
Number of vertices, edges, etc : 12, 36, 18, 6, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6,2,3}*864, {4,6,2,6}*864
   3-fold quotients : {4,6,2,2}*576
   4-fold quotients : {4,6,2,3}*432
   6-fold quotients : {4,6,2,2}*288
   9-fold quotients : {4,2,2,6}*192
   18-fold quotients : {4,2,2,3}*96, {2,2,2,6}*96
   27-fold quotients : {4,2,2,2}*64
   36-fold quotients : {2,2,2,3}*48
   54-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)(45,51)
(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);;
s1 := ( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)(10,46)
(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)(21,61)
(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)(32,68)
(33,71)(34,66)(35,69)(36,72);;
s2 := ( 1,29)( 2,28)( 3,30)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)(10,20)
(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(37,65)(38,64)(39,66)
(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,56)(47,55)(48,57)(49,62)(50,61)
(51,63)(52,59)(53,58)(54,60);;
s3 := (75,76)(77,78);;
s4 := (73,77)(74,75)(76,78);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(78)!( 4, 7)( 5, 8)( 6, 9)(13,16)(14,17)(15,18)(22,25)(23,26)(24,27)
(31,34)(32,35)(33,36)(37,46)(38,47)(39,48)(40,52)(41,53)(42,54)(43,49)(44,50)
(45,51)(55,64)(56,65)(57,66)(58,70)(59,71)(60,72)(61,67)(62,68)(63,69);
s1 := Sym(78)!( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)
(10,46)(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)
(21,61)(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)
(32,68)(33,71)(34,66)(35,69)(36,72);
s2 := Sym(78)!( 1,29)( 2,28)( 3,30)( 4,35)( 5,34)( 6,36)( 7,32)( 8,31)( 9,33)
(10,20)(11,19)(12,21)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24)(37,65)(38,64)
(39,66)(40,71)(41,70)(42,72)(43,68)(44,67)(45,69)(46,56)(47,55)(48,57)(49,62)
(50,61)(51,63)(52,59)(53,58)(54,60);
s3 := Sym(78)!(75,76)(77,78);
s4 := Sym(78)!(73,77)(74,75)(76,78);
poly := sub<Sym(78)|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*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 

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