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Polytope of Type {27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27}*54
Also Known As : 27-gon, {27}. if this polytope has another name.
Group : SmallGroup(54,1)
Rank : 2
Schlafli Type : {27}
Number of vertices, edges, etc : 27, 27
Order of s0s1 : 27
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {27,2} of size 108
   {27,4} of size 216
   {27,6} of size 324
   {27,4} of size 432
   {27,8} of size 864
   {27,18} of size 972
   {27,6} of size 972
   {27,6} of size 972
   {27,6} of size 972
   {27,6} of size 1296
   {27,12} of size 1296
   {27,8} of size 1728
Vertex Figure Of :
   {2,27} of size 108
   {4,27} of size 216
   {6,27} of size 324
   {4,27} of size 432
   {8,27} of size 864
   {18,27} of size 972
   {6,27} of size 972
   {6,27} of size 972
   {6,27} of size 972
   {6,27} of size 1296
   {12,27} of size 1296
   {8,27} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9}*18
   9-fold quotients : {3}*6
Covers (Minimal Covers in Boldface) :
   2-fold covers : {54}*108
   3-fold covers : {81}*162
   4-fold covers : {108}*216
   5-fold covers : {135}*270
   6-fold covers : {162}*324
   7-fold covers : {189}*378
   8-fold covers : {216}*432
   9-fold covers : {243}*486
   10-fold covers : {270}*540
   11-fold covers : {297}*594
   12-fold covers : {324}*648
   13-fold covers : {351}*702
   14-fold covers : {378}*756
   15-fold covers : {405}*810
   16-fold covers : {432}*864
   17-fold covers : {459}*918
   18-fold covers : {486}*972
   19-fold covers : {513}*1026
   20-fold covers : {540}*1080
   21-fold covers : {567}*1134
   22-fold covers : {594}*1188
   23-fold covers : {621}*1242
   24-fold covers : {648}*1296
   25-fold covers : {675}*1350
   26-fold covers : {702}*1404
   27-fold covers : {729}*1458
   28-fold covers : {756}*1512
   29-fold covers : {783}*1566
   30-fold covers : {810}*1620
   31-fold covers : {837}*1674
   32-fold covers : {864}*1728
   33-fold covers : {891}*1782
   34-fold covers : {918}*1836
   35-fold covers : {945}*1890
   36-fold covers : {972}*1944
   37-fold covers : {999}*1998
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)
(22,23)(24,25)(26,27);;
s1 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)
(21,22)(23,24)(25,26);;
poly := Group([s0,s1]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;;  s1 := F.2;;  
rels := [ s0*s0, s1*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(27)!( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)(12,13)(14,15)(16,17)(18,19)
(20,21)(22,23)(24,25)(26,27);
s1 := Sym(27)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)
(19,20)(21,22)(23,24)(25,26);
poly := sub<Sym(27)|s0,s1>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*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 >; 
 
References : None.
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