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# Polytope of Type {2,40}

Atlas Canonical Name : {2,40}*160
if this polytope has a name.
Group : SmallGroup(160,124)
Rank : 3
Schlafli Type : {2,40}
Number of vertices, edges, etc : 2, 40, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 2
Special Properties :
Degenerate
Universal
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,40,2} of size 320
{2,40,4} of size 640
{2,40,4} of size 640
{2,40,6} of size 960
{2,40,4} of size 1280
{2,40,8} of size 1280
{2,40,8} of size 1280
{2,40,8} of size 1280
{2,40,8} of size 1280
{2,40,4} of size 1280
{2,40,10} of size 1600
{2,40,10} of size 1600
{2,40,10} of size 1600
{2,40,12} of size 1920
{2,40,12} of size 1920
{2,40,6} of size 1920
{2,40,6} of size 1920
{2,40,6} of size 1920
{2,40,6} of size 1920
{2,40,6} of size 1920
{2,40,10} of size 1920
{2,40,10} of size 1920
Vertex Figure Of :
{2,2,40} of size 320
{3,2,40} of size 480
{4,2,40} of size 640
{5,2,40} of size 800
{6,2,40} of size 960
{7,2,40} of size 1120
{8,2,40} of size 1280
{9,2,40} of size 1440
{10,2,40} of size 1600
{11,2,40} of size 1760
{12,2,40} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,20}*80
4-fold quotients : {2,10}*40
5-fold quotients : {2,8}*32
8-fold quotients : {2,5}*20
10-fold quotients : {2,4}*16
20-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,40}*320a, {2,80}*320
3-fold covers : {6,40}*480, {2,120}*480
4-fold covers : {4,40}*640a, {8,40}*640a, {8,40}*640b, {4,80}*640a, {4,80}*640b, {2,160}*640
5-fold covers : {2,200}*800, {10,40}*800a, {10,40}*800b
6-fold covers : {6,80}*960, {12,40}*960a, {4,120}*960a, {2,240}*960
7-fold covers : {14,40}*1120, {2,280}*1120
8-fold covers : {8,40}*1280a, {4,40}*1280a, {8,40}*1280d, {4,80}*1280a, {4,80}*1280b, {16,40}*1280a, {16,40}*1280b, {8,80}*1280c, {8,80}*1280d, {16,40}*1280d, {8,80}*1280e, {8,80}*1280f, {16,40}*1280f, {4,160}*1280a, {4,160}*1280b, {2,320}*1280
9-fold covers : {18,40}*1440, {2,360}*1440, {6,120}*1440a, {6,120}*1440b, {6,120}*1440c, {6,40}*1440
10-fold covers : {4,200}*1600a, {2,400}*1600, {10,80}*1600a, {10,80}*1600b, {20,40}*1600c, {20,40}*1600d
11-fold covers : {22,40}*1760, {2,440}*1760
12-fold covers : {4,120}*1920a, {12,40}*1920a, {8,120}*1920b, {8,120}*1920c, {24,40}*1920a, {24,40}*1920c, {4,240}*1920a, {12,80}*1920a, {4,240}*1920b, {12,80}*1920b, {2,480}*1920, {6,160}*1920, {6,40}*1920d, {6,120}*1920a, {4,120}*1920c
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8,11)( 9,13)(10,12)(14,15)(16,21)(17,23)(18,22)(19,25)
(20,24)(27,32)(28,31)(29,34)(30,33)(35,36)(37,40)(38,39)(41,42);;
s2 := ( 3, 9)( 4, 6)( 5,17)( 7,19)( 8,12)(10,14)(11,27)(13,29)(15,20)(16,22)
(18,24)(21,35)(23,37)(25,30)(26,31)(28,33)(32,41)(34,38)(36,39)(40,42);;
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*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(42)!(1,2);
s1 := Sym(42)!( 4, 5)( 6, 7)( 8,11)( 9,13)(10,12)(14,15)(16,21)(17,23)(18,22)
(19,25)(20,24)(27,32)(28,31)(29,34)(30,33)(35,36)(37,40)(38,39)(41,42);
s2 := Sym(42)!( 3, 9)( 4, 6)( 5,17)( 7,19)( 8,12)(10,14)(11,27)(13,29)(15,20)
(16,22)(18,24)(21,35)(23,37)(25,30)(26,31)(28,33)(32,41)(34,38)(36,39)(40,42);
poly := sub<Sym(42)|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*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