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# Polytope of Type {2,2,5,2}

Atlas Canonical Name : {2,2,5,2}*80
if this polytope has a name.
Group : SmallGroup(80,51)
Rank : 5
Schlafli Type : {2,2,5,2}
Number of vertices, edges, etc : 2, 2, 5, 5, 2
Order of s0s1s2s3s4 : 10
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,2,5,2,2} of size 160
{2,2,5,2,3} of size 240
{2,2,5,2,4} of size 320
{2,2,5,2,5} of size 400
{2,2,5,2,6} of size 480
{2,2,5,2,7} of size 560
{2,2,5,2,8} of size 640
{2,2,5,2,9} of size 720
{2,2,5,2,10} of size 800
{2,2,5,2,11} of size 880
{2,2,5,2,12} of size 960
{2,2,5,2,13} of size 1040
{2,2,5,2,14} of size 1120
{2,2,5,2,15} of size 1200
{2,2,5,2,16} of size 1280
{2,2,5,2,17} of size 1360
{2,2,5,2,18} of size 1440
{2,2,5,2,19} of size 1520
{2,2,5,2,20} of size 1600
{2,2,5,2,21} of size 1680
{2,2,5,2,22} of size 1760
{2,2,5,2,23} of size 1840
{2,2,5,2,24} of size 1920
{2,2,5,2,25} of size 2000
Vertex Figure Of :
{2,2,2,5,2} of size 160
{3,2,2,5,2} of size 240
{4,2,2,5,2} of size 320
{5,2,2,5,2} of size 400
{6,2,2,5,2} of size 480
{7,2,2,5,2} of size 560
{8,2,2,5,2} of size 640
{9,2,2,5,2} of size 720
{10,2,2,5,2} of size 800
{11,2,2,5,2} of size 880
{12,2,2,5,2} of size 960
{13,2,2,5,2} of size 1040
{14,2,2,5,2} of size 1120
{15,2,2,5,2} of size 1200
{16,2,2,5,2} of size 1280
{17,2,2,5,2} of size 1360
{18,2,2,5,2} of size 1440
{19,2,2,5,2} of size 1520
{20,2,2,5,2} of size 1600
{21,2,2,5,2} of size 1680
{22,2,2,5,2} of size 1760
{23,2,2,5,2} of size 1840
{24,2,2,5,2} of size 1920
{25,2,2,5,2} of size 2000
Quotients (Maximal Quotients in Boldface) :
No Regular Quotients.
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,2,5,2}*160, {2,2,10,2}*160
3-fold covers : {6,2,5,2}*240, {2,2,15,2}*240
4-fold covers : {8,2,5,2}*320, {2,2,20,2}*320, {2,2,10,4}*320, {2,4,10,2}*320, {4,2,10,2}*320
5-fold covers : {2,2,25,2}*400, {2,2,5,10}*400, {2,10,5,2}*400, {10,2,5,2}*400
6-fold covers : {12,2,5,2}*480, {4,2,15,2}*480, {2,2,10,6}*480, {2,6,10,2}*480, {6,2,10,2}*480, {2,2,30,2}*480
7-fold covers : {14,2,5,2}*560, {2,2,35,2}*560
8-fold covers : {16,2,5,2}*640, {2,2,20,4}*640, {2,4,20,2}*640, {4,2,20,2}*640, {4,4,10,2}*640, {2,4,10,4}*640, {4,2,10,4}*640, {2,2,40,2}*640, {2,2,10,8}*640, {2,8,10,2}*640, {8,2,10,2}*640
9-fold covers : {18,2,5,2}*720, {2,2,45,2}*720, {2,2,15,6}*720, {2,6,15,2}*720, {6,2,15,2}*720
10-fold covers : {4,2,25,2}*800, {2,2,50,2}*800, {20,2,5,2}*800, {4,2,5,10}*800, {4,10,5,2}*800, {2,2,10,10}*800a, {2,2,10,10}*800c, {2,10,10,2}*800a, {2,10,10,2}*800b, {10,2,10,2}*800
11-fold covers : {22,2,5,2}*880, {2,2,55,2}*880
12-fold covers : {24,2,5,2}*960, {8,2,15,2}*960, {2,2,10,12}*960, {2,12,10,2}*960, {12,2,10,2}*960, {2,2,20,6}*960a, {2,6,20,2}*960a, {6,2,20,2}*960, {2,4,10,6}*960, {2,6,10,4}*960, {4,2,10,6}*960, {4,6,10,2}*960a, {6,2,10,4}*960, {6,4,10,2}*960, {2,2,60,2}*960, {2,2,30,4}*960a, {2,4,30,2}*960a, {4,2,30,2}*960, {2,2,15,6}*960, {2,6,15,2}*960, {2,2,15,4}*960, {2,4,15,2}*960
13-fold covers : {26,2,5,2}*1040, {2,2,65,2}*1040
14-fold covers : {28,2,5,2}*1120, {4,2,35,2}*1120, {2,2,10,14}*1120, {2,14,10,2}*1120, {14,2,10,2}*1120, {2,2,70,2}*1120
15-fold covers : {6,2,25,2}*1200, {2,2,75,2}*1200, {6,2,5,10}*1200, {6,10,5,2}*1200, {2,2,15,10}*1200, {2,10,15,2}*1200, {10,2,15,2}*1200, {30,2,5,2}*1200
16-fold covers : {32,2,5,2}*1280, {4,4,20,2}*1280, {2,4,20,4}*1280, {4,4,10,4}*1280, {4,2,20,4}*1280, {4,8,10,2}*1280a, {8,4,10,2}*1280a, {2,2,20,8}*1280a, {2,8,20,2}*1280a, {2,2,40,4}*1280a, {2,4,40,2}*1280a, {4,8,10,2}*1280b, {8,4,10,2}*1280b, {2,2,20,8}*1280b, {2,8,20,2}*1280b, {2,2,40,4}*1280b, {2,4,40,2}*1280b, {4,4,10,2}*1280, {2,2,20,4}*1280, {2,4,20,2}*1280, {4,2,10,8}*1280, {8,2,10,4}*1280, {2,4,10,8}*1280, {2,8,10,4}*1280, {8,2,20,2}*1280, {4,2,40,2}*1280, {2,2,10,16}*1280, {2,16,10,2}*1280, {16,2,10,2}*1280, {2,2,80,2}*1280, {2,2,5,4}*1280, {2,4,5,2}*1280
17-fold covers : {34,2,5,2}*1360, {2,2,85,2}*1360
18-fold covers : {36,2,5,2}*1440, {4,2,45,2}*1440, {2,2,10,18}*1440, {2,18,10,2}*1440, {18,2,10,2}*1440, {2,2,90,2}*1440, {12,2,15,2}*1440, {4,2,15,6}*1440, {4,6,15,2}*1440, {2,2,30,6}*1440a, {2,6,10,6}*1440, {2,6,30,2}*1440a, {6,2,10,6}*1440, {6,6,10,2}*1440a, {6,6,10,2}*1440b, {6,6,10,2}*1440c, {2,2,30,6}*1440b, {2,2,30,6}*1440c, {2,6,30,2}*1440b, {2,6,30,2}*1440c, {6,2,30,2}*1440
19-fold covers : {38,2,5,2}*1520, {2,2,95,2}*1520
20-fold covers : {8,2,25,2}*1600, {2,2,100,2}*1600, {2,2,50,4}*1600, {2,4,50,2}*1600, {4,2,50,2}*1600, {40,2,5,2}*1600, {8,2,5,10}*1600, {8,10,5,2}*1600, {2,2,10,20}*1600a, {2,2,20,10}*1600a, {2,2,20,10}*1600b, {2,10,20,2}*1600a, {2,10,20,2}*1600b, {2,20,10,2}*1600a, {10,2,20,2}*1600, {20,2,10,2}*1600, {2,4,10,10}*1600a, {2,4,10,10}*1600b, {2,10,10,4}*1600a, {2,10,10,4}*1600b, {4,2,10,10}*1600a, {4,2,10,10}*1600c, {4,10,10,2}*1600a, {10,2,10,4}*1600, {10,4,10,2}*1600, {2,2,10,20}*1600c, {2,20,10,2}*1600c, {4,10,10,2}*1600c
21-fold covers : {14,2,15,2}*1680, {42,2,5,2}*1680, {6,2,35,2}*1680, {2,2,105,2}*1680
22-fold covers : {44,2,5,2}*1760, {4,2,55,2}*1760, {2,2,10,22}*1760, {2,22,10,2}*1760, {22,2,10,2}*1760, {2,2,110,2}*1760
23-fold covers : {46,2,5,2}*1840, {2,2,115,2}*1840
24-fold covers : {16,2,15,2}*1920, {48,2,5,2}*1920, {4,4,30,2}*1920, {2,2,60,4}*1920a, {2,4,60,2}*1920a, {4,4,10,6}*1920, {4,12,10,2}*1920a, {12,4,10,2}*1920, {2,4,20,6}*1920, {2,6,20,4}*1920, {6,2,20,4}*1920, {6,4,20,2}*1920, {2,2,20,12}*1920, {2,12,20,2}*1920, {4,2,30,4}*1920a, {2,4,30,4}*1920a, {4,2,60,2}*1920, {4,6,10,4}*1920a, {6,4,10,4}*1920, {4,2,10,12}*1920, {12,2,10,4}*1920, {4,2,20,6}*1920a, {2,4,10,12}*1920, {2,12,10,4}*1920, {4,6,20,2}*1920a, {12,2,20,2}*1920, {2,2,30,8}*1920, {2,8,30,2}*1920, {8,2,30,2}*1920, {2,2,120,2}*1920, {2,6,10,8}*1920, {2,8,10,6}*1920, {6,2,10,8}*1920, {6,8,10,2}*1920, {8,2,10,6}*1920, {8,6,10,2}*1920, {2,2,10,24}*1920, {2,24,10,2}*1920, {24,2,10,2}*1920, {2,2,40,6}*1920, {2,6,40,2}*1920, {6,2,40,2}*1920, {4,2,15,6}*1920, {2,2,15,12}*1920, {2,12,15,2}*1920, {4,6,15,2}*1920, {4,2,15,4}*1920, {4,4,15,2}*1920b, {2,2,15,8}*1920, {2,8,15,2}*1920, {2,2,20,6}*1920a, {2,2,30,6}*1920, {2,6,20,2}*1920a, {2,6,30,2}*1920, {4,6,10,2}*1920a, {6,4,10,2}*1920, {6,6,10,2}*1920, {2,2,30,4}*1920, {2,4,30,2}*1920
25-fold covers : {2,2,125,2}*2000, {2,2,25,10}*2000, {2,10,25,2}*2000, {10,2,25,2}*2000, {50,2,5,2}*2000, {2,2,5,10}*2000, {2,10,5,2}*2000, {10,10,5,2}*2000a, {2,10,5,10}*2000, {10,2,5,10}*2000, {10,10,5,2}*2000b
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (3,4);;
s2 := (6,7)(8,9);;
s3 := (5,6)(7,8);;
s4 := (10,11);;
poly := Group([s0,s1,s2,s3,s4]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s1*s0*s1,
s0*s2*s0*s2, s1*s2*s1*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(11)!(1,2);
s1 := Sym(11)!(3,4);
s2 := Sym(11)!(6,7)(8,9);
s3 := Sym(11)!(5,6)(7,8);
s4 := Sym(11)!(10,11);
poly := sub<Sym(11)|s0,s1,s2,s3,s4>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s1*s0*s1, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope