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# Polytope of Type {4,10}

Atlas Canonical Name : {4,10}*1000
if this polytope has a name.
Group : SmallGroup(1000,92)
Rank : 3
Schlafli Type : {4,10}
Number of vertices, edges, etc : 50, 250, 125
Order of s0s1s2 : 20
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,10,2} of size 2000
Vertex Figure Of :
{2,4,10} of size 2000
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {4,10}*200
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,10}*2000a
Permutation Representation (GAP) :
```s0 := (  2,  5)(  3,  4)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)( 11, 64)
( 12, 63)( 13, 62)( 14, 61)( 15, 65)( 16, 92)( 17, 91)( 18, 95)( 19, 94)
( 20, 93)( 21,121)( 22,125)( 23,124)( 24,123)( 25,122)( 26,101)( 27,105)
( 28,104)( 29,103)( 30,102)( 36, 39)( 37, 38)( 41, 67)( 42, 66)( 43, 70)
( 44, 69)( 45, 68)( 46, 96)( 47,100)( 48, 99)( 49, 98)( 50, 97)( 51, 76)
( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56,107)( 57,106)( 58,110)( 59,109)
( 60,108)( 72, 75)( 73, 74)( 81, 82)( 83, 85)( 86,114)( 87,113)( 88,112)
( 89,111)( 90,115)(116,117)(118,120);;
s1 := (  2,  5)(  3,  4)(  6, 94)(  7, 93)(  8, 92)(  9, 91)( 10, 95)( 11, 31)
( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16,122)( 17,121)( 18,125)( 19,124)
( 20,123)( 21, 62)( 22, 61)( 23, 65)( 24, 64)( 25, 63)( 26, 71)( 27, 75)
( 28, 74)( 29, 73)( 30, 72)( 36, 80)( 37, 79)( 38, 78)( 39, 77)( 40, 76)
( 41, 43)( 44, 45)( 46,110)( 47,109)( 48,108)( 49,107)( 50,106)( 51,119)
( 52,118)( 53,117)( 54,116)( 55,120)( 57, 60)( 58, 59)( 66, 87)( 67, 86)
( 68, 90)( 69, 89)( 70, 88)( 81,104)( 82,103)( 83,102)( 84,101)( 85,105)
( 96,100)( 97, 99)(111,115)(112,114);;
s2 := (  1, 71)(  2, 72)(  3, 73)(  4, 74)(  5, 75)(  6, 66)(  7, 67)(  8, 68)
(  9, 69)( 10, 70)( 11, 61)( 12, 62)( 13, 63)( 14, 64)( 15, 65)( 16, 56)
( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 51)( 22, 52)( 23, 53)( 24, 54)
( 25, 55)( 26, 46)( 27, 47)( 28, 48)( 29, 49)( 30, 50)( 31, 41)( 32, 42)
( 33, 43)( 34, 44)( 35, 45)( 76,121)( 77,122)( 78,123)( 79,124)( 80,125)
( 81,116)( 82,117)( 83,118)( 84,119)( 85,120)( 86,111)( 87,112)( 88,113)
( 89,114)( 90,115)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)( 96,101)
( 97,102)( 98,103)( 99,104)(100,105);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(125)!(  2,  5)(  3,  4)(  6, 32)(  7, 31)(  8, 35)(  9, 34)( 10, 33)
( 11, 64)( 12, 63)( 13, 62)( 14, 61)( 15, 65)( 16, 92)( 17, 91)( 18, 95)
( 19, 94)( 20, 93)( 21,121)( 22,125)( 23,124)( 24,123)( 25,122)( 26,101)
( 27,105)( 28,104)( 29,103)( 30,102)( 36, 39)( 37, 38)( 41, 67)( 42, 66)
( 43, 70)( 44, 69)( 45, 68)( 46, 96)( 47,100)( 48, 99)( 49, 98)( 50, 97)
( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 56,107)( 57,106)( 58,110)
( 59,109)( 60,108)( 72, 75)( 73, 74)( 81, 82)( 83, 85)( 86,114)( 87,113)
( 88,112)( 89,111)( 90,115)(116,117)(118,120);
s1 := Sym(125)!(  2,  5)(  3,  4)(  6, 94)(  7, 93)(  8, 92)(  9, 91)( 10, 95)
( 11, 31)( 12, 35)( 13, 34)( 14, 33)( 15, 32)( 16,122)( 17,121)( 18,125)
( 19,124)( 20,123)( 21, 62)( 22, 61)( 23, 65)( 24, 64)( 25, 63)( 26, 71)
( 27, 75)( 28, 74)( 29, 73)( 30, 72)( 36, 80)( 37, 79)( 38, 78)( 39, 77)
( 40, 76)( 41, 43)( 44, 45)( 46,110)( 47,109)( 48,108)( 49,107)( 50,106)
( 51,119)( 52,118)( 53,117)( 54,116)( 55,120)( 57, 60)( 58, 59)( 66, 87)
( 67, 86)( 68, 90)( 69, 89)( 70, 88)( 81,104)( 82,103)( 83,102)( 84,101)
( 85,105)( 96,100)( 97, 99)(111,115)(112,114);
s2 := Sym(125)!(  1, 71)(  2, 72)(  3, 73)(  4, 74)(  5, 75)(  6, 66)(  7, 67)
(  8, 68)(  9, 69)( 10, 70)( 11, 61)( 12, 62)( 13, 63)( 14, 64)( 15, 65)
( 16, 56)( 17, 57)( 18, 58)( 19, 59)( 20, 60)( 21, 51)( 22, 52)( 23, 53)
( 24, 54)( 25, 55)( 26, 46)( 27, 47)( 28, 48)( 29, 49)( 30, 50)( 31, 41)
( 32, 42)( 33, 43)( 34, 44)( 35, 45)( 76,121)( 77,122)( 78,123)( 79,124)
( 80,125)( 81,116)( 82,117)( 83,118)( 84,119)( 85,120)( 86,111)( 87,112)
( 88,113)( 89,114)( 90,115)( 91,106)( 92,107)( 93,108)( 94,109)( 95,110)
( 96,101)( 97,102)( 98,103)( 99,104)(100,105);
poly := sub<Sym(125)|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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1 >;

```
References : None.
to this polytope