# Polytope of Type {4,10}

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