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# Polytope of Type {20,4}

Atlas Canonical Name : {20,4}*640a
if this polytope has a name.
Group : SmallGroup(640,2005)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 80, 160, 16
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{20,4,2} of size 1280
Vertex Figure Of :
{2,20,4} of size 1280
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4}*320
4-fold quotients : {20,4}*160
5-fold quotients : {4,4}*128
8-fold quotients : {20,2}*80, {10,4}*80
10-fold quotients : {4,4}*64
16-fold quotients : {10,2}*40
20-fold quotients : {4,4}*32
32-fold quotients : {5,2}*20
40-fold quotients : {2,4}*16, {4,2}*16
80-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {20,8}*1280a, {40,4}*1280a, {20,4}*1280a, {20,8}*1280c, {40,4}*1280c
3-fold covers : {60,4}*1920a, {20,12}*1920a
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)
(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(42,45)(43,44)(47,50)
(48,49)(52,55)(53,54)(57,60)(58,59)(61,76)(62,80)(63,79)(64,78)(65,77)(66,71)
(67,75)(68,74)(69,73)(70,72);;
s1 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,17)(12,16)(13,20)(14,19)(15,18)(21,22)
(23,25)(26,27)(28,30)(31,37)(32,36)(33,40)(34,39)(35,38)(41,62)(42,61)(43,65)
(44,64)(45,63)(46,67)(47,66)(48,70)(49,69)(50,68)(51,77)(52,76)(53,80)(54,79)
(55,78)(56,72)(57,71)(58,75)(59,74)(60,73);;
s2 := ( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)(10,50)
(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)
(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)
(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(80)!( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(21,36)
(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(42,45)(43,44)
(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(61,76)(62,80)(63,79)(64,78)(65,77)
(66,71)(67,75)(68,74)(69,73)(70,72);
s1 := Sym(80)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,17)(12,16)(13,20)(14,19)(15,18)
(21,22)(23,25)(26,27)(28,30)(31,37)(32,36)(33,40)(34,39)(35,38)(41,62)(42,61)
(43,65)(44,64)(45,63)(46,67)(47,66)(48,70)(49,69)(50,68)(51,77)(52,76)(53,80)
(54,79)(55,78)(56,72)(57,71)(58,75)(59,74)(60,73);
s2 := Sym(80)!( 1,41)( 2,42)( 3,43)( 4,44)( 5,45)( 6,46)( 7,47)( 8,48)( 9,49)
(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)
(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)
(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80);
poly := sub<Sym(80)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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