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

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

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

```

to this polytope