Polytope of Type {72,2,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,2,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,15813)
Rank : 4
Schlafli Type : {72,2,6}
Number of vertices, edges, etc : 72, 72, 6, 6
Order of s₀s₁s₂s₃ : 72
Order of s₀s₁s₂s₃s₂s₁ : 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 : {72,2,3}*864, {36,2,6}*864
   3-fold quotients : {72,2,2}*576, {24,2,6}*576
   4-fold quotients : {36,2,3}*432, {18,2,6}*432
   6-fold quotients : {36,2,2}*288, {24,2,3}*288, {12,2,6}*288
   8-fold quotients : {9,2,6}*216, {18,2,3}*216
   9-fold quotients : {24,2,2}*192, {8,2,6}*192
   12-fold quotients : {18,2,2}*144, {12,2,3}*144, {6,2,6}*144
   16-fold quotients : {9,2,3}*108
   18-fold quotients : {8,2,3}*96, {12,2,2}*96, {4,2,6}*96
   24-fold quotients : {9,2,2}*72, {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {8,2,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(19,28)(20,30)
(21,29)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33)(37,55)(38,57)(39,56)(40,62)
(41,61)(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)(51,72)
(52,68)(53,67)(54,69);;
s1 := ( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)(10,49)
(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,67)(20,69)(21,68)
(22,64)(23,66)(24,65)(25,71)(26,70)(27,72)(28,58)(29,60)(30,59)(31,55)(32,57)
(33,56)(34,62)(35,61)(36,63);;
s2 := (75,76)(77,78);;
s3 := (73,77)(74,75)(76,78);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(78)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(19,28)
(20,30)(21,29)(22,35)(23,34)(24,36)(25,32)(26,31)(27,33)(37,55)(38,57)(39,56)
(40,62)(41,61)(42,63)(43,59)(44,58)(45,60)(46,64)(47,66)(48,65)(49,71)(50,70)
(51,72)(52,68)(53,67)(54,69);
s1 := Sym(78)!( 1,40)( 2,42)( 3,41)( 4,37)( 5,39)( 6,38)( 7,44)( 8,43)( 9,45)
(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,53)(17,52)(18,54)(19,67)(20,69)
(21,68)(22,64)(23,66)(24,65)(25,71)(26,70)(27,72)(28,58)(29,60)(30,59)(31,55)
(32,57)(33,56)(34,62)(35,61)(36,63);
s2 := Sym(78)!(75,76)(77,78);
s3 := Sym(78)!(73,77)(74,75)(76,78);
poly := sub<Sym(78)|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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

to this polytope