Polytope of Type {2,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,27}*108
if this polytope has a name.
Group : SmallGroup(108,4)
Rank : 3
Schlafli Type : {2,27}
Number of vertices, edges, etc : 2, 27, 27
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 :
   {2,27,2} of size 216
   {2,27,4} of size 432
   {2,27,6} of size 648
   {2,27,4} of size 864
   {2,27,8} of size 1728
   {2,27,18} of size 1944
   {2,27,6} of size 1944
   {2,27,6} of size 1944
   {2,27,6} of size 1944
Vertex Figure Of :
   {2,2,27} of size 216
   {3,2,27} of size 324
   {4,2,27} of size 432
   {5,2,27} of size 540
   {6,2,27} of size 648
   {7,2,27} of size 756
   {8,2,27} of size 864
   {9,2,27} of size 972
   {10,2,27} of size 1080
   {11,2,27} of size 1188
   {12,2,27} of size 1296
   {13,2,27} of size 1404
   {14,2,27} of size 1512
   {15,2,27} of size 1620
   {16,2,27} of size 1728
   {17,2,27} of size 1836
   {18,2,27} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,9}*36
   9-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,54}*216
   3-fold covers : {2,81}*324, {6,27}*324
   4-fold covers : {2,108}*432, {4,54}*432a, {4,27}*432
   5-fold covers : {2,135}*540
   6-fold covers : {2,162}*648, {6,54}*648a, {6,54}*648b
   7-fold covers : {2,189}*756
   8-fold covers : {4,108}*864a, {2,216}*864, {8,54}*864, {8,27}*864, {4,54}*864
   9-fold covers : {2,243}*972, {18,27}*972, {6,27}*972a, {6,81}*972
   10-fold covers : {10,54}*1080, {2,270}*1080
   11-fold covers : {2,297}*1188
   12-fold covers : {2,324}*1296, {4,162}*1296a, {4,81}*1296, {12,54}*1296a, {6,108}*1296a, {6,108}*1296b, {12,54}*1296b, {6,27}*1296, {12,27}*1296
   13-fold covers : {2,351}*1404
   14-fold covers : {14,54}*1512, {2,378}*1512
   15-fold covers : {2,405}*1620, {6,135}*1620
   16-fold covers : {4,216}*1728a, {4,108}*1728a, {4,216}*1728b, {8,108}*1728a, {8,108}*1728b, {2,432}*1728, {16,54}*1728, {8,27}*1728, {4,108}*1728b, {4,54}*1728b, {4,108}*1728c, {8,54}*1728b, {8,54}*1728c
   17-fold covers : {2,459}*1836
   18-fold covers : {2,486}*1944, {18,54}*1944a, {18,54}*1944b, {6,54}*1944a, {6,54}*1944b, {6,162}*1944a, {6,162}*1944b, {6,54}*1944g
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 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);;
s2 := ( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)
(23,24)(25,26)(27,28);;
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*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope