# Polytope of Type {25,10}

Atlas Canonical Name : {25,10}*500
if this polytope has a name.
Group : SmallGroup(500,26)
Rank : 3
Schlafli Type : {25,10}
Number of vertices, edges, etc : 25, 125, 10
Order of s0s1s2 : 50
Order of s0s1s2s1 : 10
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{25,10,2} of size 1000
{25,10,4} of size 2000
Vertex Figure Of :
{2,25,10} of size 1000
Quotients (Maximal Quotients in Boldface) :
5-fold quotients : {25,2}*100, {5,10}*100
25-fold quotients : {5,2}*20
Covers (Minimal Covers in Boldface) :
2-fold covers : {50,10}*1000b
3-fold covers : {75,10}*1500
4-fold covers : {100,10}*2000b, {50,20}*2000b
Permutation Representation (GAP) :
s0 := (  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)( 11, 16)
( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,102)( 27,101)( 28,105)( 29,104)
( 30,103)( 31,122)( 32,121)( 33,125)( 34,124)( 35,123)( 36,117)( 37,116)
( 38,120)( 39,119)( 40,118)( 41,112)( 42,111)( 43,115)( 44,114)( 45,113)
( 46,107)( 47,106)( 48,110)( 49,109)( 50,108)( 51, 77)( 52, 76)( 53, 80)
( 54, 79)( 55, 78)( 56, 97)( 57, 96)( 58,100)( 59, 99)( 60, 98)( 61, 92)
( 62, 91)( 63, 95)( 64, 94)( 65, 93)( 66, 87)( 67, 86)( 68, 90)( 69, 89)
( 70, 88)( 71, 82)( 72, 81)( 73, 85)( 74, 84)( 75, 83);;
s1 := (  1, 31)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 26)(  7, 30)(  8, 29)
(  9, 28)( 10, 27)( 11, 46)( 12, 50)( 13, 49)( 14, 48)( 15, 47)( 16, 41)
( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 36)( 22, 40)( 23, 39)( 24, 38)
( 25, 37)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,102)( 57,101)
( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)( 64,124)( 65,123)
( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)( 72,111)( 73,115)
( 74,114)( 75,113)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)( 86, 97)
( 87, 96)( 88,100)( 89, 99)( 90, 98)( 91, 92)( 93, 95);;
s2 := (  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)( 13, 18)
( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)( 36, 41)
( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)( 59, 74)
( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)( 82, 97)
( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)( 90, 95)
(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)(113,118)
(114,119)(115,120);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(125)!(  2,  5)(  3,  4)(  6, 21)(  7, 25)(  8, 24)(  9, 23)( 10, 22)
( 11, 16)( 12, 20)( 13, 19)( 14, 18)( 15, 17)( 26,102)( 27,101)( 28,105)
( 29,104)( 30,103)( 31,122)( 32,121)( 33,125)( 34,124)( 35,123)( 36,117)
( 37,116)( 38,120)( 39,119)( 40,118)( 41,112)( 42,111)( 43,115)( 44,114)
( 45,113)( 46,107)( 47,106)( 48,110)( 49,109)( 50,108)( 51, 77)( 52, 76)
( 53, 80)( 54, 79)( 55, 78)( 56, 97)( 57, 96)( 58,100)( 59, 99)( 60, 98)
( 61, 92)( 62, 91)( 63, 95)( 64, 94)( 65, 93)( 66, 87)( 67, 86)( 68, 90)
( 69, 89)( 70, 88)( 71, 82)( 72, 81)( 73, 85)( 74, 84)( 75, 83);
s1 := Sym(125)!(  1, 31)(  2, 35)(  3, 34)(  4, 33)(  5, 32)(  6, 26)(  7, 30)
(  8, 29)(  9, 28)( 10, 27)( 11, 46)( 12, 50)( 13, 49)( 14, 48)( 15, 47)
( 16, 41)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21, 36)( 22, 40)( 23, 39)
( 24, 38)( 25, 37)( 51,107)( 52,106)( 53,110)( 54,109)( 55,108)( 56,102)
( 57,101)( 58,105)( 59,104)( 60,103)( 61,122)( 62,121)( 63,125)( 64,124)
( 65,123)( 66,117)( 67,116)( 68,120)( 69,119)( 70,118)( 71,112)( 72,111)
( 73,115)( 74,114)( 75,113)( 76, 82)( 77, 81)( 78, 85)( 79, 84)( 80, 83)
( 86, 97)( 87, 96)( 88,100)( 89, 99)( 90, 98)( 91, 92)( 93, 95);
s2 := Sym(125)!(  6, 21)(  7, 22)(  8, 23)(  9, 24)( 10, 25)( 11, 16)( 12, 17)
( 13, 18)( 14, 19)( 15, 20)( 31, 46)( 32, 47)( 33, 48)( 34, 49)( 35, 50)
( 36, 41)( 37, 42)( 38, 43)( 39, 44)( 40, 45)( 56, 71)( 57, 72)( 58, 73)
( 59, 74)( 60, 75)( 61, 66)( 62, 67)( 63, 68)( 64, 69)( 65, 70)( 81, 96)
( 82, 97)( 83, 98)( 84, 99)( 85,100)( 86, 91)( 87, 92)( 88, 93)( 89, 94)
( 90, 95)(106,121)(107,122)(108,123)(109,124)(110,125)(111,116)(112,117)
(113,118)(114,119)(115,120);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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