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# Polytope of Type {2,4,20}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,20}*640
if this polytope has a name.
Group : SmallGroup(640,12497)
Rank : 4
Schlafli Type : {2,4,20}
Number of vertices, edges, etc : 2, 8, 80, 40
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,20,2} of size 1280
Vertex Figure Of :
{2,2,4,20} of size 1280
{3,2,4,20} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,20}*320
4-fold quotients : {2,2,20}*160, {2,4,10}*160
5-fold quotients : {2,4,4}*128
8-fold quotients : {2,2,10}*80
10-fold quotients : {2,4,4}*64
16-fold quotients : {2,2,5}*40
20-fold quotients : {2,2,4}*32, {2,4,2}*32
40-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,8,20}*1280a, {2,4,40}*1280a, {4,4,20}*1280b, {2,4,20}*1280a, {2,4,40}*1280b, {2,8,20}*1280b
3-fold covers : {2,4,60}*1920a, {6,4,20}*1920a, {2,12,20}*1920a
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (23,28)(24,29)(25,30)(26,31)(27,32)(33,38)(34,39)(35,40)(36,41)(37,42)
(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)(63,78)
(64,79)(65,80)(66,81)(67,82)(68,73)(69,74)(70,75)(71,76)(72,77);;
s2 := ( 3,43)( 4,47)( 5,46)( 6,45)( 7,44)( 8,48)( 9,52)(10,51)(11,50)(12,49)
(13,53)(14,57)(15,56)(16,55)(17,54)(18,58)(19,62)(20,61)(21,60)(22,59)(23,63)
(24,67)(25,66)(26,65)(27,64)(28,68)(29,72)(30,71)(31,70)(32,69)(33,73)(34,77)
(35,76)(36,75)(37,74)(38,78)(39,82)(40,81)(41,80)(42,79);;
s3 := ( 3, 4)( 5, 7)( 8, 9)(10,12)(13,19)(14,18)(15,22)(16,21)(17,20)(23,24)
(25,27)(28,29)(30,32)(33,39)(34,38)(35,42)(36,41)(37,40)(43,64)(44,63)(45,67)
(46,66)(47,65)(48,69)(49,68)(50,72)(51,71)(52,70)(53,79)(54,78)(55,82)(56,81)
(57,80)(58,74)(59,73)(60,77)(61,76)(62,75);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(82)!(1,2);
s1 := Sym(82)!(23,28)(24,29)(25,30)(26,31)(27,32)(33,38)(34,39)(35,40)(36,41)
(37,42)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(51,61)(52,62)
(63,78)(64,79)(65,80)(66,81)(67,82)(68,73)(69,74)(70,75)(71,76)(72,77);
s2 := Sym(82)!( 3,43)( 4,47)( 5,46)( 6,45)( 7,44)( 8,48)( 9,52)(10,51)(11,50)
(12,49)(13,53)(14,57)(15,56)(16,55)(17,54)(18,58)(19,62)(20,61)(21,60)(22,59)
(23,63)(24,67)(25,66)(26,65)(27,64)(28,68)(29,72)(30,71)(31,70)(32,69)(33,73)
(34,77)(35,76)(36,75)(37,74)(38,78)(39,82)(40,81)(41,80)(42,79);
s3 := Sym(82)!( 3, 4)( 5, 7)( 8, 9)(10,12)(13,19)(14,18)(15,22)(16,21)(17,20)
(23,24)(25,27)(28,29)(30,32)(33,39)(34,38)(35,42)(36,41)(37,40)(43,64)(44,63)
(45,67)(46,66)(47,65)(48,69)(49,68)(50,72)(51,71)(52,70)(53,79)(54,78)(55,82)
(56,81)(57,80)(58,74)(59,73)(60,77)(61,76)(62,75);
poly := sub<Sym(82)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s1*s3*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```

to this polytope