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# Polytope of Type {4,10}

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

```
References : None.
to this polytope