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# Polytope of Type {2,10,2}

Atlas Canonical Name : {2,10,2}*80
if this polytope has a name.
Group : SmallGroup(80,51)
Rank : 4
Schlafli Type : {2,10,2}
Number of vertices, edges, etc : 2, 10, 10, 2
Order of s0s1s2s3 : 10
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,10,2,2} of size 160
{2,10,2,3} of size 240
{2,10,2,4} of size 320
{2,10,2,5} of size 400
{2,10,2,6} of size 480
{2,10,2,7} of size 560
{2,10,2,8} of size 640
{2,10,2,9} of size 720
{2,10,2,10} of size 800
{2,10,2,11} of size 880
{2,10,2,12} of size 960
{2,10,2,13} of size 1040
{2,10,2,14} of size 1120
{2,10,2,15} of size 1200
{2,10,2,16} of size 1280
{2,10,2,17} of size 1360
{2,10,2,18} of size 1440
{2,10,2,19} of size 1520
{2,10,2,20} of size 1600
{2,10,2,21} of size 1680
{2,10,2,22} of size 1760
{2,10,2,23} of size 1840
{2,10,2,24} of size 1920
{2,10,2,25} of size 2000
Vertex Figure Of :
{2,2,10,2} of size 160
{3,2,10,2} of size 240
{4,2,10,2} of size 320
{5,2,10,2} of size 400
{6,2,10,2} of size 480
{7,2,10,2} of size 560
{8,2,10,2} of size 640
{9,2,10,2} of size 720
{10,2,10,2} of size 800
{11,2,10,2} of size 880
{12,2,10,2} of size 960
{13,2,10,2} of size 1040
{14,2,10,2} of size 1120
{15,2,10,2} of size 1200
{16,2,10,2} of size 1280
{17,2,10,2} of size 1360
{18,2,10,2} of size 1440
{19,2,10,2} of size 1520
{20,2,10,2} of size 1600
{21,2,10,2} of size 1680
{22,2,10,2} of size 1760
{23,2,10,2} of size 1840
{24,2,10,2} of size 1920
{25,2,10,2} of size 2000
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,5,2}*40
5-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,20,2}*160, {2,10,4}*160, {4,10,2}*160
3-fold covers : {2,10,6}*240, {6,10,2}*240, {2,30,2}*240
4-fold covers : {2,20,4}*320, {4,20,2}*320, {4,10,4}*320, {2,40,2}*320, {2,10,8}*320, {8,10,2}*320
5-fold covers : {2,50,2}*400, {2,10,10}*400a, {2,10,10}*400c, {10,10,2}*400a, {10,10,2}*400b
6-fold covers : {2,10,12}*480, {12,10,2}*480, {2,20,6}*480a, {6,20,2}*480a, {4,10,6}*480, {6,10,4}*480, {2,60,2}*480, {2,30,4}*480a, {4,30,2}*480a
7-fold covers : {2,10,14}*560, {14,10,2}*560, {2,70,2}*560
8-fold covers : {4,20,4}*640, {2,40,4}*640a, {4,40,2}*640a, {2,20,4}*640, {4,20,2}*640, {2,40,4}*640b, {4,40,2}*640b, {2,20,8}*640a, {8,20,2}*640a, {2,20,8}*640b, {8,20,2}*640b, {4,10,8}*640, {8,10,4}*640, {2,80,2}*640, {2,10,16}*640, {16,10,2}*640
9-fold covers : {2,10,18}*720, {18,10,2}*720, {2,90,2}*720, {6,10,6}*720, {2,30,6}*720a, {6,30,2}*720a, {2,30,6}*720b, {2,30,6}*720c, {6,30,2}*720b, {6,30,2}*720c
10-fold covers : {2,100,2}*800, {2,50,4}*800, {4,50,2}*800, {2,10,20}*800a, {2,20,10}*800a, {2,20,10}*800b, {10,20,2}*800a, {10,20,2}*800b, {20,10,2}*800a, {4,10,10}*800a, {4,10,10}*800b, {10,10,4}*800a, {10,10,4}*800b, {2,10,20}*800c, {20,10,2}*800c
11-fold covers : {2,10,22}*880, {22,10,2}*880, {2,110,2}*880
12-fold covers : {4,20,6}*960, {6,20,4}*960, {4,10,12}*960, {12,10,4}*960, {2,10,24}*960, {24,10,2}*960, {2,40,6}*960, {6,40,2}*960, {6,10,8}*960, {8,10,6}*960, {2,20,12}*960, {12,20,2}*960, {2,60,4}*960a, {4,60,2}*960a, {4,30,4}*960a, {2,120,2}*960, {2,30,8}*960, {8,30,2}*960, {2,20,6}*960c, {2,30,6}*960, {6,20,2}*960c, {6,30,2}*960, {2,30,4}*960, {4,30,2}*960
13-fold covers : {2,10,26}*1040, {26,10,2}*1040, {2,130,2}*1040
14-fold covers : {2,20,14}*1120, {14,20,2}*1120, {2,10,28}*1120, {28,10,2}*1120, {4,10,14}*1120, {14,10,4}*1120, {2,140,2}*1120, {2,70,4}*1120, {4,70,2}*1120
15-fold covers : {2,50,6}*1200, {6,50,2}*1200, {2,150,2}*1200, {6,10,10}*1200a, {6,10,10}*1200c, {10,10,6}*1200a, {10,10,6}*1200b, {2,10,30}*1200a, {30,10,2}*1200a, {2,10,30}*1200b, {2,30,10}*1200b, {2,30,10}*1200c, {10,30,2}*1200b, {10,30,2}*1200c, {30,10,2}*1200b
16-fold covers : {2,20,8}*1280a, {8,20,2}*1280a, {2,40,4}*1280a, {4,40,2}*1280a, {2,40,8}*1280a, {8,40,2}*1280a, {2,40,8}*1280b, {2,40,8}*1280c, {8,40,2}*1280b, {8,40,2}*1280c, {2,40,8}*1280d, {8,40,2}*1280d, {8,10,8}*1280, {4,20,8}*1280a, {8,20,4}*1280a, {4,20,8}*1280b, {8,20,4}*1280b, {4,40,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,40,4}*1280d, {2,20,16}*1280a, {16,20,2}*1280a, {2,80,4}*1280a, {4,80,2}*1280a, {2,20,16}*1280b, {16,20,2}*1280b, {2,80,4}*1280b, {4,80,2}*1280b, {2,20,4}*1280a, {2,40,4}*1280b, {4,20,2}*1280a, {4,40,2}*1280b, {2,20,8}*1280b, {8,20,2}*1280b, {4,10,16}*1280, {16,10,4}*1280, {2,10,32}*1280, {32,10,2}*1280, {2,160,2}*1280, {2,10,4}*1280b, {4,10,2}*1280b
17-fold covers : {2,10,34}*1360, {34,10,2}*1360, {2,170,2}*1360
18-fold covers : {2,10,36}*1440, {36,10,2}*1440, {2,20,18}*1440a, {18,20,2}*1440a, {4,10,18}*1440, {18,10,4}*1440, {2,180,2}*1440, {2,90,4}*1440a, {4,90,2}*1440a, {6,10,12}*1440, {12,10,6}*1440, {6,20,6}*1440, {2,60,6}*1440a, {6,60,2}*1440a, {2,30,12}*1440a, {12,30,2}*1440a, {4,30,6}*1440a, {6,30,4}*1440a, {2,30,12}*1440b, {12,30,2}*1440b, {2,60,6}*1440b, {2,60,6}*1440c, {6,60,2}*1440b, {6,60,2}*1440c, {4,30,6}*1440b, {4,30,6}*1440c, {6,30,4}*1440b, {6,30,4}*1440c, {2,30,12}*1440c, {12,30,2}*1440c, {2,20,4}*1440, {2,30,4}*1440, {4,20,2}*1440, {4,30,2}*1440, {2,20,6}*1440, {6,20,2}*1440
19-fold covers : {2,10,38}*1520, {38,10,2}*1520, {2,190,2}*1520
20-fold covers : {2,100,4}*1600, {4,100,2}*1600, {4,50,4}*1600, {2,200,2}*1600, {2,50,8}*1600, {8,50,2}*1600, {4,10,20}*1600a, {20,10,4}*1600a, {4,20,10}*1600a, {4,20,10}*1600b, {10,20,4}*1600a, {10,20,4}*1600b, {2,10,40}*1600a, {2,40,10}*1600a, {2,40,10}*1600b, {10,40,2}*1600a, {10,40,2}*1600b, {40,10,2}*1600a, {8,10,10}*1600a, {8,10,10}*1600b, {10,10,8}*1600a, {10,10,8}*1600b, {2,20,20}*1600a, {2,20,20}*1600c, {20,20,2}*1600a, {20,20,2}*1600b, {4,10,20}*1600c, {20,10,4}*1600c, {2,10,40}*1600c, {40,10,2}*1600c
21-fold covers : {6,10,14}*1680, {14,10,6}*1680, {2,30,14}*1680, {14,30,2}*1680, {2,10,42}*1680, {42,10,2}*1680, {2,70,6}*1680, {6,70,2}*1680, {2,210,2}*1680
22-fold covers : {2,20,22}*1760, {22,20,2}*1760, {2,10,44}*1760, {44,10,2}*1760, {4,10,22}*1760, {22,10,4}*1760, {2,220,2}*1760, {2,110,4}*1760, {4,110,2}*1760
23-fold covers : {2,10,46}*1840, {46,10,2}*1840, {2,230,2}*1840
24-fold covers : {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*1920, {2,60,8}*1920a, {8,60,2}*1920a, {2,120,4}*1920a, {4,120,2}*1920a, {6,20,8}*1920a, {8,20,6}*1920a, {4,40,6}*1920a, {6,40,4}*1920a, {2,40,12}*1920a, {12,40,2}*1920a, {2,20,24}*1920a, {24,20,2}*1920a, {2,60,8}*1920b, {8,60,2}*1920b, {2,120,4}*1920b, {4,120,2}*1920b, {6,20,8}*1920b, {8,20,6}*1920b, {4,40,6}*1920b, {6,40,4}*1920b, {2,40,12}*1920b, {12,40,2}*1920b, {2,20,24}*1920b, {24,20,2}*1920b, {2,60,4}*1920a, {4,60,2}*1920a, {4,20,6}*1920a, {6,20,4}*1920a, {2,20,12}*1920a, {12,20,2}*1920a, {4,30,8}*1920a, {8,30,4}*1920a, {8,10,12}*1920, {12,10,8}*1920, {4,10,24}*1920, {24,10,4}*1920, {2,30,16}*1920, {16,30,2}*1920, {2,240,2}*1920, {6,10,16}*1920, {16,10,6}*1920, {2,10,48}*1920, {48,10,2}*1920, {2,80,6}*1920, {6,80,2}*1920, {2,20,12}*1920b, {12,20,2}*1920b, {2,20,6}*1920a, {2,60,6}*1920a, {6,20,2}*1920a, {6,60,2}*1920a, {4,20,6}*1920c, {4,30,6}*1920, {6,20,4}*1920c, {6,30,4}*1920, {2,30,12}*1920a, {12,30,2}*1920a, {2,30,6}*1920, {2,40,6}*1920b, {6,30,2}*1920, {6,40,2}*1920b, {2,40,6}*1920c, {2,60,6}*1920b, {6,40,2}*1920c, {6,60,2}*1920b, {2,20,12}*1920c, {2,30,12}*1920b, {12,20,2}*1920c, {12,30,2}*1920b, {2,60,4}*1920b, {4,60,2}*1920b, {4,30,4}*1920a, {4,30,4}*1920b, {2,30,4}*1920b, {2,60,4}*1920c, {4,30,2}*1920b, {4,60,2}*1920c, {2,30,8}*1920b, {8,30,2}*1920b, {2,30,8}*1920c, {8,30,2}*1920c
25-fold covers : {2,250,2}*2000, {2,10,50}*2000a, {2,50,10}*2000a, {2,50,10}*2000b, {10,50,2}*2000a, {10,50,2}*2000b, {50,10,2}*2000a, {2,10,10}*2000a, {2,10,10}*2000c, {10,10,2}*2000b, {10,10,2}*2000c, {10,10,10}*2000b, {10,10,10}*2000d, {10,10,10}*2000e, {10,10,10}*2000f, {2,10,10}*2000d, {10,10,2}*2000d
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := ( 5, 6)( 7, 8)( 9,10)(11,12);;
s2 := ( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,12);;
s3 := (13,14);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3, 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(14)!(1,2);
s1 := Sym(14)!( 5, 6)( 7, 8)( 9,10)(11,12);
s2 := Sym(14)!( 3, 7)( 4, 5)( 6,11)( 8, 9)(10,12);
s3 := Sym(14)!(13,14);
poly := sub<Sym(14)|s0,s1,s2,s3>;

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

```

to this polytope