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# Polytope of Type {4,10,4}

Atlas Canonical Name : {4,10,4}*320
Also Known As : {{4,10|2},{10,4|2}}. if this polytope has another name.
Group : SmallGroup(320,1262)
Rank : 4
Schlafli Type : {4,10,4}
Number of vertices, edges, etc : 4, 20, 20, 4
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,10,4,2} of size 640
{4,10,4,4} of size 1280
{4,10,4,6} of size 1920
{4,10,4,3} of size 1920
Vertex Figure Of :
{2,4,10,4} of size 640
{4,4,10,4} of size 1280
{6,4,10,4} of size 1920
{3,4,10,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,10,4}*160, {4,10,2}*160
4-fold quotients : {2,10,2}*80
5-fold quotients : {4,2,4}*64
8-fold quotients : {2,5,2}*40
10-fold quotients : {2,2,4}*32, {4,2,2}*32
20-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,20,4}*640, {4,10,8}*640, {8,10,4}*640
3-fold covers : {4,10,12}*960, {12,10,4}*960, {4,30,4}*960a
4-fold covers : {8,10,8}*1280, {4,20,8}*1280a, {8,20,4}*1280a, {4,20,8}*1280b, {8,20,4}*1280b, {4,40,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,40,4}*1280d, {4,10,16}*1280, {16,10,4}*1280
5-fold covers : {4,50,4}*1600, {4,10,20}*1600a, {20,10,4}*1600a, {4,10,20}*1600c, {20,10,4}*1600c
6-fold covers : {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*1920, {4,30,8}*1920a, {8,30,4}*1920a, {8,10,12}*1920, {12,10,8}*1920, {4,10,24}*1920, {24,10,4}*1920
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)(21,71)
(22,75)(23,74)(24,73)(25,72)(26,76)(27,80)(28,79)(29,78)(30,77)(31,61)(32,65)
(33,64)(34,63)(35,62)(36,66)(37,70)(38,69)(39,68)(40,67);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,37)(22,36)
(23,40)(24,39)(25,38)(26,32)(27,31)(28,35)(29,34)(30,33)(41,42)(43,45)(46,47)
(48,50)(51,52)(53,55)(56,57)(58,60)(61,77)(62,76)(63,80)(64,79)(65,78)(66,72)
(67,71)(68,75)(69,74)(70,73);;
s3 := ( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)(10,30)
(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,71)
(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)(52,62)
(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70);;
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*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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)
(21,71)(22,75)(23,74)(24,73)(25,72)(26,76)(27,80)(28,79)(29,78)(30,77)(31,61)
(32,65)(33,64)(34,63)(35,62)(36,66)(37,70)(38,69)(39,68)(40,67);
s2 := Sym(80)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,37)
(22,36)(23,40)(24,39)(25,38)(26,32)(27,31)(28,35)(29,34)(30,33)(41,42)(43,45)
(46,47)(48,50)(51,52)(53,55)(56,57)(58,60)(61,77)(62,76)(63,80)(64,79)(65,78)
(66,72)(67,71)(68,75)(69,74)(70,73);
s3 := Sym(80)!( 1,21)( 2,22)( 3,23)( 4,24)( 5,25)( 6,26)( 7,27)( 8,28)( 9,29)
(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)
(41,71)(42,72)(43,73)(44,74)(45,75)(46,76)(47,77)(48,78)(49,79)(50,80)(51,61)
(52,62)(53,63)(54,64)(55,65)(56,66)(57,67)(58,68)(59,69)(60,70);
poly := sub<Sym(80)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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