Polytope of Type {27,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {27,2}*108
if this polytope has a name.
Group : SmallGroup(108,4)
Rank : 3
Schlafli Type : {27,2}
Number of vertices, edges, etc : 27, 27, 2
Order of s₀s₁s₂ : 54
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {27,2,2} of size 216
   {27,2,3} of size 324
   {27,2,4} of size 432
   {27,2,5} of size 540
   {27,2,6} of size 648
   {27,2,7} of size 756
   {27,2,8} of size 864
   {27,2,9} of size 972
   {27,2,10} of size 1080
   {27,2,11} of size 1188
   {27,2,12} of size 1296
   {27,2,13} of size 1404
   {27,2,14} of size 1512
   {27,2,15} of size 1620
   {27,2,16} of size 1728
   {27,2,17} of size 1836
   {27,2,18} of size 1944
Vertex Figure Of :
   {2,27,2} of size 216
   {4,27,2} of size 432
   {6,27,2} of size 648
   {4,27,2} of size 864
   {8,27,2} of size 1728
   {18,27,2} of size 1944
   {6,27,2} of size 1944
   {6,27,2} of size 1944
   {6,27,2} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {9,2}*36
   9-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {54,2}*216
   3-fold covers : {81,2}*324, {27,6}*324
   4-fold covers : {108,2}*432, {54,4}*432a, {27,4}*432
   5-fold covers : {135,2}*540
   6-fold covers : {162,2}*648, {54,6}*648a, {54,6}*648b
   7-fold covers : {189,2}*756
   8-fold covers : {108,4}*864a, {216,2}*864, {54,8}*864, {27,8}*864, {54,4}*864
   9-fold covers : {243,2}*972, {27,18}*972, {27,6}*972a, {81,6}*972
   10-fold covers : {54,10}*1080, {270,2}*1080
   11-fold covers : {297,2}*1188
   12-fold covers : {324,2}*1296, {162,4}*1296a, {81,4}*1296, {54,12}*1296a, {108,6}*1296a, {108,6}*1296b, {54,12}*1296b, {27,6}*1296, {27,12}*1296
   13-fold covers : {351,2}*1404
   14-fold covers : {54,14}*1512, {378,2}*1512
   15-fold covers : {405,2}*1620, {135,6}*1620
   16-fold covers : {216,4}*1728a, {108,4}*1728a, {216,4}*1728b, {108,8}*1728a, {108,8}*1728b, {432,2}*1728, {54,16}*1728, {27,8}*1728, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c
   17-fold covers : {459,2}*1836
   18-fold covers : {486,2}*1944, {54,18}*1944a, {54,18}*1944b, {54,6}*1944a, {54,6}*1944b, {162,6}*1944a, {162,6}*1944b, {54,6}*1944g
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);;
s2 := (28,29);;
poly := Group([s0,s1,s2]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2, 
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(29)!( 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(29)!( 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);
s2 := Sym(29)!(28,29);
poly := sub<Sym(29)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2, 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