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# Polytope of Type {40,4}

Atlas Canonical Name : {40,4}*320a
Also Known As : {40,4|2}. if this polytope has another name.
Group : SmallGroup(320,359)
Rank : 3
Schlafli Type : {40,4}
Number of vertices, edges, etc : 40, 80, 4
Order of s0s1s2 : 40
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{40,4,2} of size 640
{40,4,4} of size 1280
{40,4,6} of size 1920
{40,4,3} of size 1920
Vertex Figure Of :
{2,40,4} of size 640
{4,40,4} of size 1280
{4,40,4} of size 1280
{6,40,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {20,4}*160, {40,2}*160
4-fold quotients : {20,2}*80, {10,4}*80
5-fold quotients : {8,4}*64a
8-fold quotients : {10,2}*40
10-fold quotients : {4,4}*32, {8,2}*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, {40,8}*640a, {40,8}*640b, {80,4}*640a, {80,4}*640b
3-fold covers : {40,12}*960a, {120,4}*960a
4-fold covers : {40,8}*1280a, {40,4}*1280a, {40,8}*1280d, {80,4}*1280a, {80,4}*1280b, {40,16}*1280a, {40,16}*1280b, {80,8}*1280c, {80,8}*1280d, {40,16}*1280d, {80,8}*1280e, {80,8}*1280f, {40,16}*1280f, {160,4}*1280a, {160,4}*1280b
5-fold covers : {200,4}*1600a, {40,20}*1600c, {40,20}*1600d
6-fold covers : {120,4}*1920a, {40,12}*1920a, {120,8}*1920b, {120,8}*1920c, {40,24}*1920a, {40,24}*1920c, {240,4}*1920a, {80,12}*1920a, {240,4}*1920b, {80,12}*1920b
Permutation Representation (GAP) :
```s0 := ( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)
(23,29)(24,28)(25,27)(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)(51,71)(52,75)(53,74)(54,73)
(55,72)(56,76)(57,80)(58,79)(59,78)(60,77);;
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)(21,67)
(22,66)(23,70)(24,69)(25,68)(26,62)(27,61)(28,65)(29,64)(30,63)(31,77)(32,76)
(33,80)(34,79)(35,78)(36,72)(37,71)(38,75)(39,74)(40,73);;
s2 := (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)
(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80);;
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, 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*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*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)(21,26)
(22,30)(23,29)(24,28)(25,27)(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)(51,71)(52,75)(53,74)
(54,73)(55,72)(56,76)(57,80)(58,79)(59,78)(60,77);
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)
(21,67)(22,66)(23,70)(24,69)(25,68)(26,62)(27,61)(28,65)(29,64)(30,63)(31,77)
(32,76)(33,80)(34,79)(35,78)(36,72)(37,71)(38,75)(39,74)(40,73);
s2 := Sym(80)!(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)
(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80);
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, 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*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope