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# Polytope of Type {10,10}

Atlas Canonical Name : {10,10}*200a
Also Known As : {10,10|2}. if this polytope has another name.
Group : SmallGroup(200,49)
Rank : 3
Schlafli Type : {10,10}
Number of vertices, edges, etc : 10, 50, 10
Order of s0s1s2 : 10
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{10,10,2} of size 400
{10,10,4} of size 800
{10,10,5} of size 1000
{10,10,3} of size 1200
{10,10,5} of size 1200
{10,10,6} of size 1200
{10,10,8} of size 1600
{10,10,10} of size 2000
{10,10,10} of size 2000
{10,10,10} of size 2000
Vertex Figure Of :
{2,10,10} of size 400
{4,10,10} of size 800
{5,10,10} of size 1000
{3,10,10} of size 1200
{5,10,10} of size 1200
{6,10,10} of size 1200
{8,10,10} of size 1600
{10,10,10} of size 2000
{10,10,10} of size 2000
{10,10,10} of size 2000
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {2,10}*40, {10,2}*40
10-fold quotients : {2,5}*20, {5,2}*20
25-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {10,20}*400a, {20,10}*400a
3-fold covers : {10,30}*600b, {30,10}*600b
4-fold covers : {10,40}*800a, {40,10}*800a, {20,20}*800a
5-fold covers : {10,50}*1000a, {50,10}*1000a, {10,10}*1000c, {10,10}*1000d
6-fold covers : {20,30}*1200b, {30,20}*1200b, {10,60}*1200b, {60,10}*1200b
7-fold covers : {10,70}*1400b, {70,10}*1400b
8-fold covers : {10,80}*1600a, {80,10}*1600a, {20,20}*1600a, {20,40}*1600c, {40,20}*1600c, {20,40}*1600e, {40,20}*1600e
9-fold covers : {10,90}*1800b, {90,10}*1800b, {30,30}*1800a, {30,30}*1800c, {30,30}*1800g
10-fold covers : {20,50}*2000a, {50,20}*2000a, {10,100}*2000a, {100,10}*2000a, {10,20}*2000b, {20,10}*2000b, {10,20}*2000h, {20,10}*2000h
Permutation Representation (GAP) :
```s0 := (  2,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)( 18, 19)
( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)( 38, 39)
( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)( 58, 59)
( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)( 78, 79)
( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)( 98, 99);;
s1 := (  1, 52)(  2, 51)(  3, 55)(  4, 54)(  5, 53)(  6, 72)(  7, 71)(  8, 75)
(  9, 74)( 10, 73)( 11, 67)( 12, 66)( 13, 70)( 14, 69)( 15, 68)( 16, 62)
( 17, 61)( 18, 65)( 19, 64)( 20, 63)( 21, 57)( 22, 56)( 23, 60)( 24, 59)
( 25, 58)( 26, 77)( 27, 76)( 28, 80)( 29, 79)( 30, 78)( 31, 97)( 32, 96)
( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)( 40, 93)
( 41, 87)( 42, 86)( 43, 90)( 44, 89)( 45, 88)( 46, 82)( 47, 81)( 48, 85)
( 49, 84)( 50, 83);;
s2 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)(  8, 78)
(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)( 16, 91)
( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)( 24, 89)
( 25, 90)( 26, 56)( 27, 57)( 28, 58)( 29, 59)( 30, 60)( 31, 51)( 32, 52)
( 33, 53)( 34, 54)( 35, 55)( 36, 71)( 37, 72)( 38, 73)( 39, 74)( 40, 75)
( 41, 66)( 42, 67)( 43, 68)( 44, 69)( 45, 70)( 46, 61)( 47, 62)( 48, 63)
( 49, 64)( 50, 65);;
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*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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,  5)(  3,  4)(  7, 10)(  8,  9)( 12, 15)( 13, 14)( 17, 20)
( 18, 19)( 22, 25)( 23, 24)( 27, 30)( 28, 29)( 32, 35)( 33, 34)( 37, 40)
( 38, 39)( 42, 45)( 43, 44)( 47, 50)( 48, 49)( 52, 55)( 53, 54)( 57, 60)
( 58, 59)( 62, 65)( 63, 64)( 67, 70)( 68, 69)( 72, 75)( 73, 74)( 77, 80)
( 78, 79)( 82, 85)( 83, 84)( 87, 90)( 88, 89)( 92, 95)( 93, 94)( 97,100)
( 98, 99);
s1 := Sym(100)!(  1, 52)(  2, 51)(  3, 55)(  4, 54)(  5, 53)(  6, 72)(  7, 71)
(  8, 75)(  9, 74)( 10, 73)( 11, 67)( 12, 66)( 13, 70)( 14, 69)( 15, 68)
( 16, 62)( 17, 61)( 18, 65)( 19, 64)( 20, 63)( 21, 57)( 22, 56)( 23, 60)
( 24, 59)( 25, 58)( 26, 77)( 27, 76)( 28, 80)( 29, 79)( 30, 78)( 31, 97)
( 32, 96)( 33,100)( 34, 99)( 35, 98)( 36, 92)( 37, 91)( 38, 95)( 39, 94)
( 40, 93)( 41, 87)( 42, 86)( 43, 90)( 44, 89)( 45, 88)( 46, 82)( 47, 81)
( 48, 85)( 49, 84)( 50, 83);
s2 := Sym(100)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 76)(  7, 77)
(  8, 78)(  9, 79)( 10, 80)( 11, 96)( 12, 97)( 13, 98)( 14, 99)( 15,100)
( 16, 91)( 17, 92)( 18, 93)( 19, 94)( 20, 95)( 21, 86)( 22, 87)( 23, 88)
( 24, 89)( 25, 90)( 26, 56)( 27, 57)( 28, 58)( 29, 59)( 30, 60)( 31, 51)
( 32, 52)( 33, 53)( 34, 54)( 35, 55)( 36, 71)( 37, 72)( 38, 73)( 39, 74)
( 40, 75)( 41, 66)( 42, 67)( 43, 68)( 44, 69)( 45, 70)( 46, 61)( 47, 62)
( 48, 63)( 49, 64)( 50, 65);
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*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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