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# Polytope of Type {4,40}

Atlas Canonical Name : {4,40}*320b
if this polytope has a name.
Group : SmallGroup(320,374)
Rank : 3
Schlafli Type : {4,40}
Number of vertices, edges, etc : 4, 80, 40
Order of s0s1s2 : 40
Order of s0s1s2s1 : 4
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
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*160
4-fold quotients : {2,20}*80, {4,10}*80
5-fold quotients : {4,8}*64b
8-fold quotients : {2,10}*40
10-fold quotients : {4,4}*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}*640c, {8,40}*640d
3-fold covers : {12,40}*960b, {4,120}*960b
4-fold covers : {8,40}*1280a, {4,40}*1280a, {8,40}*1280d, {4,80}*1280a, {4,80}*1280b, {8,80}*1280a, {8,80}*1280b, {16,40}*1280c, {16,40}*1280e
5-fold covers : {4,200}*1600b, {20,40}*1600e, {20,40}*1600f
6-fold covers : {4,120}*1920a, {12,40}*1920a, {8,120}*1920a, {24,40}*1920b, {8,120}*1920d, {24,40}*1920d
Permutation Representation (GAP) :
```s0 := (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);;
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)(11,17)(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)(41,77)(42,76)(43,80)
(44,79)(45,78)(46,72)(47,71)(48,75)(49,74)(50,73)(51,62)(52,61)(53,65)(54,64)
(55,63)(56,67)(57,66)(58,70)(59,69)(60,68);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := 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);
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)(11,17)(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)(41,77)(42,76)
(43,80)(44,79)(45,78)(46,72)(47,71)(48,75)(49,74)(50,73)(51,62)(52,61)(53,65)
(54,64)(55,63)(56,67)(57,66)(58,70)(59,69)(60,68);
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,
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1,
s0*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*s0*s1*s2*s1 >;

```
References : None.
to this polytope