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# Polytope of Type {20,4}

Atlas Canonical Name : {20,4}*320
if this polytope has a name.
Group : SmallGroup(320,368)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 40, 80, 8
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{20,4,2} of size 640
{20,4,4} of size 1280
{20,4,6} of size 1920
{20,4,3} of size 1920
Vertex Figure Of :
{2,20,4} of size 640
{4,20,4} of size 1280
{6,20,4} of size 1920
{6,20,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4}*160
4-fold quotients : {20,2}*80, {10,4}*80
5-fold quotients : {4,4}*64
8-fold quotients : {10,2}*40
10-fold quotients : {4,4}*32
16-fold quotients : {5,2}*20
20-fold quotients : {2,4}*16, {4,2}*16
40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {40,4}*640a, {20,8}*640a, {20,4}*640a, {40,4}*640b, {20,8}*640b
3-fold covers : {20,12}*960a, {60,4}*960a
4-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, {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
5-fold covers : {100,4}*1600, {20,20}*1600a, {20,20}*1600c
6-fold covers : {60,8}*1920a, {120,4}*1920a, {40,12}*1920a, {20,24}*1920a, {60,4}*1920a, {120,4}*1920b, {60,8}*1920b, {40,12}*1920b, {20,24}*1920b, {20,12}*1920a
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(11,16)(12,20)(13,19)(14,18)(15,17)(22,25)
(23,24)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(41,61)(42,65)(43,64)
(44,63)(45,62)(46,66)(47,70)(48,69)(49,68)(50,67)(51,76)(52,80)(53,79)(54,78)
(55,77)(56,71)(57,75)(58,74)(59,73)(60,72);;
s1 := ( 1,42)( 2,41)( 3,45)( 4,44)( 5,43)( 6,47)( 7,46)( 8,50)( 9,49)(10,48)
(11,52)(12,51)(13,55)(14,54)(15,53)(16,57)(17,56)(18,60)(19,59)(20,58)(21,62)
(22,61)(23,65)(24,64)(25,63)(26,67)(27,66)(28,70)(29,69)(30,68)(31,72)(32,71)
(33,75)(34,74)(35,73)(36,77)(37,76)(38,80)(39,79)(40,78);;
s2 := (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)
(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)
(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75);;
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*s2*s1*s2*s1*s0*s1*s0*s2*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*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope