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# Polytope of Type {4,40}

Atlas Canonical Name : {4,40}*320a
Also Known As : {4,40|2}. if this polytope has another name.
Group : SmallGroup(320,359)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 4, 80, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,40,2} of size 640
{4,40,4} of size 1280
{4,40,4} of size 1280
{4,40,6} of size 1920
Vertex Figure Of :
{2,4,40} of size 640
{4,4,40} of size 1280
{6,4,40} of size 1920
{3,4,40} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*160, {2,40}*160
4-fold quotients : {2,20}*80, {4,10}*80
5-fold quotients : {4,8}*64a
8-fold quotients : {2,10}*40
10-fold quotients : {4,4}*32, {2,8}*32
16-fold quotients : {2,5}*20
20-fold quotients : {2,4}*16, {4,2}*16
40-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,40}*640a, {8,40}*640a, {8,40}*640b, {4,80}*640a, {4,80}*640b
3-fold covers : {12,40}*960a, {4,120}*960a
4-fold covers : {8,40}*1280a, {4,40}*1280a, {8,40}*1280d, {4,80}*1280a, {4,80}*1280b, {16,40}*1280a, {16,40}*1280b, {8,80}*1280c, {8,80}*1280d, {16,40}*1280d, {8,80}*1280e, {8,80}*1280f, {16,40}*1280f, {4,160}*1280a, {4,160}*1280b
5-fold covers : {4,200}*1600a, {20,40}*1600c, {20,40}*1600d
6-fold covers : {4,120}*1920a, {12,40}*1920a, {8,120}*1920b, {8,120}*1920c, {24,40}*1920a, {24,40}*1920c, {4,240}*1920a, {12,80}*1920a, {4,240}*1920b, {12,80}*1920b
Permutation Representation (GAP) :
```s0 := (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);;
s1 := ( 1,41)( 2,45)( 3,44)( 4,43)( 5,42)( 6,46)( 7,50)( 8,49)( 9,48)(10,47)
(11,51)(12,55)(13,54)(14,53)(15,52)(16,56)(17,60)(18,59)(19,58)(20,57)(21,66)
(22,70)(23,69)(24,68)(25,67)(26,61)(27,65)(28,64)(29,63)(30,62)(31,76)(32,80)
(33,79)(34,78)(35,77)(36,71)(37,75)(38,74)(39,73)(40,72);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,27)(22,26)
(23,30)(24,29)(25,28)(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)(51,72)(52,71)(53,75)(54,74)
(55,73)(56,77)(57,76)(58,80)(59,79)(60,78);;
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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(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);
s1 := Sym(80)!( 1,41)( 2,45)( 3,44)( 4,43)( 5,42)( 6,46)( 7,50)( 8,49)( 9,48)
(10,47)(11,51)(12,55)(13,54)(14,53)(15,52)(16,56)(17,60)(18,59)(19,58)(20,57)
(21,66)(22,70)(23,69)(24,68)(25,67)(26,61)(27,65)(28,64)(29,63)(30,62)(31,76)
(32,80)(33,79)(34,78)(35,77)(36,71)(37,75)(38,74)(39,73)(40,72);
s2 := Sym(80)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,27)
(22,26)(23,30)(24,29)(25,28)(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)(51,72)(52,71)(53,75)
(54,74)(55,73)(56,77)(57,76)(58,80)(59,79)(60,78);
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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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