Polytope of Type {2,2,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,27}*216
if this polytope has a name.
Group : SmallGroup(216,23)
Rank : 4
Schlafli Type : {2,2,27}
Number of vertices, edges, etc : 2, 2, 27, 27
Order of s₀s₁s₂s₃ : 54
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,2,27,2} of size 432
   {2,2,27,4} of size 864
   {2,2,27,6} of size 1296
   {2,2,27,4} of size 1728
Vertex Figure Of :
   {2,2,2,27} of size 432
   {3,2,2,27} of size 648
   {4,2,2,27} of size 864
   {5,2,2,27} of size 1080
   {6,2,2,27} of size 1296
   {7,2,2,27} of size 1512
   {8,2,2,27} of size 1728
   {9,2,2,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,9}*72
   9-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,2,27}*432, {2,2,54}*432
   3-fold covers : {2,2,81}*648, {2,6,27}*648, {6,2,27}*648
   4-fold covers : {8,2,27}*864, {2,2,108}*864, {2,4,54}*864a, {4,2,54}*864, {2,4,27}*864
   5-fold covers : {10,2,27}*1080, {2,2,135}*1080
   6-fold covers : {4,2,81}*1296, {2,2,162}*1296, {12,2,27}*1296, {4,6,27}*1296, {2,6,54}*1296a, {2,6,54}*1296b, {6,2,54}*1296
   7-fold covers : {14,2,27}*1512, {2,2,189}*1512
   8-fold covers : {16,2,27}*1728, {2,4,108}*1728a, {4,2,108}*1728, {4,4,54}*1728, {2,2,216}*1728, {2,8,54}*1728, {8,2,54}*1728, {4,4,27}*1728b, {2,8,27}*1728, {2,4,54}*1728
   9-fold covers : {2,2,243}*1944, {2,18,27}*1944, {18,2,27}*1944, {6,6,27}*1944a, {2,6,27}*1944a, {2,6,81}*1944, {6,2,81}*1944, {6,6,27}*1944b
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (3,4);;
s2 := ( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)
(26,27)(28,29)(30,31);;
s3 := ( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)
(25,26)(27,28)(29,30);;
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, s1*s2*s1*s2, s0*s3*s0*s3, 
s1*s3*s1*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*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(31)!(1,2);
s1 := Sym(31)!(3,4);
s2 := Sym(31)!( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)
(24,25)(26,27)(28,29)(30,31);
s3 := Sym(31)!( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28)(29,30);
poly := sub<Sym(31)|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, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

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