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# Polytope of Type {4,4,10}

Atlas Canonical Name : {4,4,10}*320
Also Known As : {{4,4|2},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(320,1260)
Rank : 4
Schlafli Type : {4,4,10}
Number of vertices, edges, etc : 4, 8, 20, 10
Order of s0s1s2s3 : 20
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,10,2} of size 640
{4,4,10,4} of size 1280
{4,4,10,5} of size 1600
{4,4,10,6} of size 1920
{4,4,10,3} of size 1920
{4,4,10,5} of size 1920
Vertex Figure Of :
{2,4,4,10} of size 640
{4,4,4,10} of size 1280
{6,4,4,10} of size 1920
{3,4,4,10} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,10}*160, {4,2,10}*160
4-fold quotients : {4,2,5}*80, {2,2,10}*80
5-fold quotients : {4,4,2}*64
8-fold quotients : {2,2,5}*40
10-fold quotients : {2,4,2}*32, {4,2,2}*32
20-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,20}*640, {4,8,10}*640a, {8,4,10}*640a, {4,8,10}*640b, {8,4,10}*640b, {4,4,10}*640
3-fold covers : {4,12,10}*960a, {12,4,10}*960, {4,4,30}*960
4-fold covers : {4,8,10}*1280a, {8,4,10}*1280a, {8,8,10}*1280a, {8,8,10}*1280b, {8,8,10}*1280c, {8,8,10}*1280d, {8,4,20}*1280a, {4,4,40}*1280a, {8,4,20}*1280b, {4,4,40}*1280b, {4,8,20}*1280a, {4,4,20}*1280a, {4,4,20}*1280b, {4,8,20}*1280b, {4,8,20}*1280c, {4,8,20}*1280d, {4,16,10}*1280a, {16,4,10}*1280a, {4,16,10}*1280b, {16,4,10}*1280b, {4,4,10}*1280, {4,8,10}*1280b, {8,4,10}*1280b
5-fold covers : {4,4,50}*1600, {4,20,10}*1600a, {20,4,10}*1600, {4,20,10}*1600c
6-fold covers : {4,4,60}*1920, {4,12,20}*1920a, {12,4,20}*1920, {4,8,30}*1920a, {8,4,30}*1920a, {8,12,10}*1920a, {12,8,10}*1920a, {4,24,10}*1920a, {24,4,10}*1920a, {4,8,30}*1920b, {8,4,30}*1920b, {8,12,10}*1920b, {12,8,10}*1920b, {4,24,10}*1920b, {24,4,10}*1920b, {4,4,30}*1920a, {4,12,10}*1920a, {12,4,10}*1920a
Permutation Representation (GAP) :
```s0 := (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)
(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80);;
s1 := ( 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)(41,61)
(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)
(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80);;
s2 := ( 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);;
s3 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)(51,52)
(53,55)(56,57)(58,60)(61,62)(63,65)(66,67)(68,70)(71,72)(73,75)(76,77)
(78,80);;
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*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80);
s1 := 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)
(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)
(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80);
s2 := 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);
s3 := Sym(80)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,32)(33,35)(36,37)(38,40)(41,42)(43,45)(46,47)(48,50)
(51,52)(53,55)(56,57)(58,60)(61,62)(63,65)(66,67)(68,70)(71,72)(73,75)(76,77)
(78,80);
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*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope