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# Polytope of Type {10,4}

Atlas Canonical Name : {10,4}*400
Also Known As : {10,4}4if this polytope has another name.
Group : SmallGroup(400,211)
Rank : 3
Schlafli Type : {10,4}
Number of vertices, edges, etc : 50, 100, 20
Order of s0s1s2 : 4
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{10,4,2} of size 800
{10,4,4} of size 1600
Vertex Figure Of :
{2,10,4} of size 800
{4,10,4} of size 1600
{5,10,4} of size 2000
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {10,4}*200
25-fold quotients : {2,4}*16
50-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,8}*800, {20,4}*800
3-fold covers : {30,4}*1200b, {10,12}*1200c
4-fold covers : {10,16}*1600, {20,4}*1600, {20,8}*1600a, {40,4}*1600a, {40,4}*1600b, {20,8}*1600b
5-fold covers : {10,4}*2000a, {10,20}*2000d, {10,20}*2000e, {10,20}*2000f, {10,20}*2000g, {10,4}*2000b, {10,20}*2000i, {10,20}*2000j
Permutation Representation (GAP) :
```s0 := (  1, 26)(  2, 30)(  3, 29)(  4, 28)(  5, 27)(  6, 46)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 41)( 12, 45)( 13, 44)( 14, 43)( 15, 42)( 16, 36)
( 17, 40)( 18, 39)( 19, 38)( 20, 37)( 21, 31)( 22, 35)( 23, 34)( 24, 33)
( 25, 32)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 96)( 57,100)
( 58, 99)( 59, 98)( 60, 97)( 61, 91)( 62, 95)( 63, 94)( 64, 93)( 65, 92)
( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)( 72, 85)( 73, 84)
( 74, 83)( 75, 82);;
s1 := ( 1, 8)( 2,18)( 4,13)( 5,23)( 7,16)( 9,11)(10,21)(12,19)(15,24)(20,22)
(26,33)(27,43)(29,38)(30,48)(32,41)(34,36)(35,46)(37,44)(40,49)(45,47)(51,58)
(52,68)(54,63)(55,73)(57,66)(59,61)(60,71)(62,69)(65,74)(70,72)(76,83)(77,93)
(79,88)(80,98)(82,91)(84,86)(85,96)(87,94)(90,99)(95,97);;
s2 := (  1, 51)(  2, 56)(  3, 61)(  4, 66)(  5, 71)(  6, 52)(  7, 57)(  8, 62)
(  9, 67)( 10, 72)( 11, 53)( 12, 58)( 13, 63)( 14, 68)( 15, 73)( 16, 54)
( 17, 59)( 18, 64)( 19, 69)( 20, 74)( 21, 55)( 22, 60)( 23, 65)( 24, 70)
( 25, 75)( 26, 76)( 27, 81)( 28, 86)( 29, 91)( 30, 96)( 31, 77)( 32, 82)
( 33, 87)( 34, 92)( 35, 97)( 36, 78)( 37, 83)( 38, 88)( 39, 93)( 40, 98)
( 41, 79)( 42, 84)( 43, 89)( 44, 94)( 45, 99)( 46, 80)( 47, 85)( 48, 90)
( 49, 95)( 50,100);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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(100)!(  1, 26)(  2, 30)(  3, 29)(  4, 28)(  5, 27)(  6, 46)(  7, 50)
(  8, 49)(  9, 48)( 10, 47)( 11, 41)( 12, 45)( 13, 44)( 14, 43)( 15, 42)
( 16, 36)( 17, 40)( 18, 39)( 19, 38)( 20, 37)( 21, 31)( 22, 35)( 23, 34)
( 24, 33)( 25, 32)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56, 96)
( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61, 91)( 62, 95)( 63, 94)( 64, 93)
( 65, 92)( 66, 86)( 67, 90)( 68, 89)( 69, 88)( 70, 87)( 71, 81)( 72, 85)
( 73, 84)( 74, 83)( 75, 82);
s1 := Sym(100)!( 1, 8)( 2,18)( 4,13)( 5,23)( 7,16)( 9,11)(10,21)(12,19)(15,24)
(20,22)(26,33)(27,43)(29,38)(30,48)(32,41)(34,36)(35,46)(37,44)(40,49)(45,47)
(51,58)(52,68)(54,63)(55,73)(57,66)(59,61)(60,71)(62,69)(65,74)(70,72)(76,83)
(77,93)(79,88)(80,98)(82,91)(84,86)(85,96)(87,94)(90,99)(95,97);
s2 := Sym(100)!(  1, 51)(  2, 56)(  3, 61)(  4, 66)(  5, 71)(  6, 52)(  7, 57)
(  8, 62)(  9, 67)( 10, 72)( 11, 53)( 12, 58)( 13, 63)( 14, 68)( 15, 73)
( 16, 54)( 17, 59)( 18, 64)( 19, 69)( 20, 74)( 21, 55)( 22, 60)( 23, 65)
( 24, 70)( 25, 75)( 26, 76)( 27, 81)( 28, 86)( 29, 91)( 30, 96)( 31, 77)
( 32, 82)( 33, 87)( 34, 92)( 35, 97)( 36, 78)( 37, 83)( 38, 88)( 39, 93)
( 40, 98)( 41, 79)( 42, 84)( 43, 89)( 44, 94)( 45, 99)( 46, 80)( 47, 85)
( 48, 90)( 49, 95)( 50,100);
poly := sub<Sym(100)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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