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# Polytope of Type {10,4}

Atlas Canonical Name : {10,4}*80
Also Known As : {10,4|2}. if this polytope has another name.
Group : SmallGroup(80,39)
Rank : 3
Schlafli Type : {10,4}
Number of vertices, edges, etc : 10, 20, 4
Order of s0s1s2 : 20
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{10,4,2} of size 160
{10,4,4} of size 320
{10,4,6} of size 480
{10,4,3} of size 480
{10,4,8} of size 640
{10,4,8} of size 640
{10,4,4} of size 640
{10,4,6} of size 720
{10,4,10} of size 800
{10,4,12} of size 960
{10,4,6} of size 960
{10,4,14} of size 1120
{10,4,5} of size 1200
{10,4,8} of size 1280
{10,4,16} of size 1280
{10,4,16} of size 1280
{10,4,4} of size 1280
{10,4,8} of size 1280
{10,4,18} of size 1440
{10,4,9} of size 1440
{10,4,4} of size 1440
{10,4,6} of size 1440
{10,4,20} of size 1600
{10,4,22} of size 1760
{10,4,24} of size 1920
{10,4,24} of size 1920
{10,4,12} of size 1920
{10,4,12} of size 1920
{10,4,6} of size 1920
{10,4,12} of size 1920
{10,4,10} of size 2000
Vertex Figure Of :
{2,10,4} of size 160
{4,10,4} of size 320
{5,10,4} of size 400
{3,10,4} of size 480
{5,10,4} of size 480
{6,10,4} of size 480
{8,10,4} of size 640
{10,10,4} of size 800
{10,10,4} of size 800
{10,10,4} of size 800
{12,10,4} of size 960
{4,10,4} of size 960
{6,10,4} of size 960
{3,10,4} of size 960
{5,10,4} of size 960
{6,10,4} of size 960
{6,10,4} of size 960
{10,10,4} of size 960
{10,10,4} of size 960
{14,10,4} of size 1120
{3,10,4} of size 1200
{15,10,4} of size 1200
{16,10,4} of size 1280
{4,10,4} of size 1280
{5,10,4} of size 1280
{18,10,4} of size 1440
{20,10,4} of size 1600
{20,10,4} of size 1600
{20,10,4} of size 1600
{4,10,4} of size 1600
{22,10,4} of size 1760
{24,10,4} of size 1920
{4,10,4} of size 1920
{6,10,4} of size 1920
{6,10,4} of size 1920
{10,10,4} of size 1920
{25,10,4} of size 2000
{5,10,4} of size 2000
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,2}*40
4-fold quotients : {5,2}*20
5-fold quotients : {2,4}*16
10-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,4}*160, {10,8}*160
3-fold covers : {10,12}*240, {30,4}*240a
4-fold covers : {40,4}*320a, {20,4}*320, {40,4}*320b, {20,8}*320a, {20,8}*320b, {10,16}*320
5-fold covers : {50,4}*400, {10,20}*400a, {10,20}*400c
6-fold covers : {10,24}*480, {20,12}*480, {60,4}*480a, {30,8}*480
7-fold covers : {10,28}*560, {70,4}*560
8-fold covers : {40,4}*640a, {40,8}*640a, {40,8}*640b, {20,8}*640a, {40,8}*640c, {40,8}*640d, {80,4}*640a, {80,4}*640b, {20,4}*640a, {40,4}*640b, {20,8}*640b, {20,16}*640a, {20,16}*640b, {10,32}*640
9-fold covers : {10,36}*720, {90,4}*720a, {30,12}*720a, {30,12}*720b, {30,12}*720c, {30,4}*720
10-fold covers : {100,4}*800, {50,8}*800, {10,40}*800a, {20,20}*800a, {20,20}*800c, {10,40}*800c
11-fold covers : {10,44}*880, {110,4}*880
12-fold covers : {10,48}*960, {20,12}*960a, {20,24}*960a, {40,12}*960a, {20,24}*960b, {40,12}*960b, {120,4}*960a, {60,4}*960a, {120,4}*960b, {60,8}*960a, {60,8}*960b, {30,16}*960, {20,12}*960b, {30,12}*960b, {30,4}*960b
13-fold covers : {10,52}*1040, {130,4}*1040
14-fold covers : {10,56}*1120, {20,28}*1120, {140,4}*1120, {70,8}*1120
15-fold covers : {50,12}*1200, {150,4}*1200a, {10,60}*1200a, {30,20}*1200b, {10,60}*1200b, {30,20}*1200c
16-fold covers : {40,8}*1280a, {20,8}*1280a, {40,8}*1280b, {40,4}*1280a, {40,8}*1280c, {40,8}*1280d, {20,16}*1280a, {80,4}*1280a, {20,16}*1280b, {80,4}*1280b, {80,8}*1280a, {40,16}*1280a, {80,8}*1280b, {40,16}*1280b, {40,16}*1280c, {80,8}*1280c, {80,8}*1280d, {40,16}*1280d, {40,16}*1280e, {80,8}*1280e, {80,8}*1280f, {40,16}*1280f, {20,32}*1280a, {160,4}*1280a, {20,32}*1280b, {160,4}*1280b, {20,4}*1280a, {40,4}*1280b, {20,8}*1280b, {20,8}*1280c, {40,8}*1280e, {40,4}*1280c, {40,4}*1280d, {20,8}*1280d, {40,8}*1280f, {40,8}*1280g, {40,8}*1280h, {10,64}*1280, {10,4}*1280a
17-fold covers : {10,68}*1360, {170,4}*1360
18-fold covers : {10,72}*1440, {20,36}*1440, {180,4}*1440a, {90,8}*1440, {30,24}*1440a, {60,12}*1440a, {30,24}*1440b, {60,12}*1440b, {60,12}*1440c, {30,24}*1440c, {20,4}*1440, {60,4}*1440, {30,8}*1440, {20,12}*1440
19-fold covers : {10,76}*1520, {190,4}*1520
20-fold covers : {200,4}*1600a, {100,4}*1600, {200,4}*1600b, {100,8}*1600a, {100,8}*1600b, {50,16}*1600, {10,80}*1600a, {20,40}*1600a, {20,20}*1600a, {20,20}*1600c, {20,40}*1600b, {20,40}*1600c, {40,20}*1600c, {40,20}*1600d, {20,40}*1600e, {40,20}*1600e, {40,20}*1600f, {10,80}*1600c
21-fold covers : {30,28}*1680a, {10,84}*1680, {70,12}*1680, {210,4}*1680a
22-fold covers : {10,88}*1760, {20,44}*1760, {220,4}*1760, {110,8}*1760
23-fold covers : {10,92}*1840, {230,4}*1840
24-fold covers : {60,8}*1920a, {120,4}*1920a, {40,12}*1920a, {20,24}*1920a, {120,8}*1920a, {120,8}*1920b, {120,8}*1920c, {40,24}*1920a, {40,24}*1920b, {40,24}*1920c, {120,8}*1920d, {40,24}*1920d, {60,16}*1920a, {240,4}*1920a, {80,12}*1920a, {20,48}*1920a, {60,16}*1920b, {240,4}*1920b, {80,12}*1920b, {20,48}*1920b, {60,4}*1920a, {120,4}*1920b, {60,8}*1920b, {40,12}*1920b, {20,24}*1920b, {20,12}*1920a, {30,32}*1920, {10,96}*1920, {40,12}*1920e, {40,12}*1920f, {20,24}*1920c, {20,24}*1920d, {20,12}*1920c, {60,12}*1920c, {30,24}*1920a, {30,12}*1920, {60,12}*1920d, {30,24}*1920b, {60,4}*1920d, {30,8}*1920f, {30,8}*1920g, {60,4}*1920e, {30,4}*1920b
25-fold covers : {250,4}*2000, {50,20}*2000a, {10,100}*2000a, {10,20}*2000b, {50,20}*2000b, {10,20}*2000c, {10,20}*2000h, {10,4}*2000b
Permutation Representation (GAP) :
```s0 := ( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)(15,16)(17,18)(19,20);;
s1 := ( 1, 3)( 2,11)( 4, 8)( 5, 6)( 7,17)(10,15)(12,13)(14,18)(16,19);;
s2 := ( 1, 2)( 3, 6)( 4, 7)( 5,10)( 8,13)( 9,14)(11,15)(12,16)(17,19)(18,20);;
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*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(20)!( 3, 4)( 6, 7)( 8, 9)(11,12)(13,14)(15,16)(17,18)(19,20);
s1 := Sym(20)!( 1, 3)( 2,11)( 4, 8)( 5, 6)( 7,17)(10,15)(12,13)(14,18)(16,19);
s2 := Sym(20)!( 1, 2)( 3, 6)( 4, 7)( 5,10)( 8,13)( 9,14)(11,15)(12,16)(17,19)
(18,20);
poly := sub<Sym(20)|s0,s1,s2>;

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

```
References : None.
to this polytope