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# Polytope of Type {8,20}

Atlas Canonical Name : {8,20}*1280k
if this polytope has a name.
Group : SmallGroup(1280,1116427)
Rank : 3
Schlafli Type : {8,20}
Number of vertices, edges, etc : 32, 320, 80
Order of s0s1s2 : 20
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,20}*640c, {8,10}*640a
4-fold quotients : {8,5}*320a, {4,10}*320a
8-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)(  8,126)
(  9,116)( 10,115)( 11,114)( 12,113)( 13,120)( 14,119)( 15,118)( 16,117)
( 17,108)( 18,107)( 19,106)( 20,105)( 21,112)( 22,111)( 23,110)( 24,109)
( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)( 32,102)
( 33, 91)( 34, 92)( 35, 89)( 36, 90)( 37, 95)( 38, 96)( 39, 93)( 40, 94)
( 41, 84)( 42, 83)( 43, 82)( 44, 81)( 45, 88)( 46, 87)( 47, 86)( 48, 85)
( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 80)( 54, 79)( 55, 78)( 56, 77)
( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 61, 71)( 62, 72)( 63, 69)( 64, 70);;
s1 := (  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 26)( 18, 25)( 19, 28)( 20, 27)
( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 33, 41)( 34, 42)( 35, 43)( 36, 44)
( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 50)( 51, 52)( 53, 56)( 54, 55)
( 57, 58)( 59, 60)( 61, 64)( 62, 63)( 65,121)( 66,122)( 67,123)( 68,124)
( 69,127)( 70,128)( 71,125)( 72,126)( 73,113)( 74,114)( 75,115)( 76,116)
( 77,119)( 78,120)( 79,117)( 80,118)( 81, 97)( 82, 98)( 83, 99)( 84,100)
( 85,103)( 86,104)( 87,101)( 88,102)( 89,105)( 90,106)( 91,107)( 92,108)
( 93,111)( 94,112)( 95,109)( 96,110);;
s2 := (  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 87)( 10, 88)( 11, 85)( 12, 86)
( 13, 83)( 14, 84)( 15, 81)( 16, 82)( 17,111)( 18,112)( 19,109)( 20,110)
( 21,107)( 22,108)( 23,105)( 24,106)( 25, 63)( 26, 64)( 27, 61)( 28, 62)
( 29, 59)( 30, 60)( 31, 57)( 32, 58)( 33, 40)( 34, 39)( 35, 38)( 36, 37)
( 41,119)( 42,120)( 43,117)( 44,118)( 45,115)( 46,116)( 47,113)( 48,114)
( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)( 56, 73)
( 65,103)( 66,104)( 67,101)( 68,102)( 69, 99)( 70,100)( 71, 97)( 72, 98)
( 89, 96)( 90, 95)( 91, 94)( 92, 93)(121,127)(122,128)(123,125)(124,126);;
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*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(128)!(  1,123)(  2,124)(  3,121)(  4,122)(  5,127)(  6,128)(  7,125)
(  8,126)(  9,116)( 10,115)( 11,114)( 12,113)( 13,120)( 14,119)( 15,118)
( 16,117)( 17,108)( 18,107)( 19,106)( 20,105)( 21,112)( 22,111)( 23,110)
( 24,109)( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)
( 32,102)( 33, 91)( 34, 92)( 35, 89)( 36, 90)( 37, 95)( 38, 96)( 39, 93)
( 40, 94)( 41, 84)( 42, 83)( 43, 82)( 44, 81)( 45, 88)( 46, 87)( 47, 86)
( 48, 85)( 49, 76)( 50, 75)( 51, 74)( 52, 73)( 53, 80)( 54, 79)( 55, 78)
( 56, 77)( 57, 67)( 58, 68)( 59, 65)( 60, 66)( 61, 71)( 62, 72)( 63, 69)
( 64, 70);
s1 := Sym(128)!(  5,  7)(  6,  8)( 13, 15)( 14, 16)( 17, 26)( 18, 25)( 19, 28)
( 20, 27)( 21, 32)( 22, 31)( 23, 30)( 24, 29)( 33, 41)( 34, 42)( 35, 43)
( 36, 44)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 50)( 51, 52)( 53, 56)
( 54, 55)( 57, 58)( 59, 60)( 61, 64)( 62, 63)( 65,121)( 66,122)( 67,123)
( 68,124)( 69,127)( 70,128)( 71,125)( 72,126)( 73,113)( 74,114)( 75,115)
( 76,116)( 77,119)( 78,120)( 79,117)( 80,118)( 81, 97)( 82, 98)( 83, 99)
( 84,100)( 85,103)( 86,104)( 87,101)( 88,102)( 89,105)( 90,106)( 91,107)
( 92,108)( 93,111)( 94,112)( 95,109)( 96,110);
s2 := Sym(128)!(  1,  7)(  2,  8)(  3,  5)(  4,  6)(  9, 87)( 10, 88)( 11, 85)
( 12, 86)( 13, 83)( 14, 84)( 15, 81)( 16, 82)( 17,111)( 18,112)( 19,109)
( 20,110)( 21,107)( 22,108)( 23,105)( 24,106)( 25, 63)( 26, 64)( 27, 61)
( 28, 62)( 29, 59)( 30, 60)( 31, 57)( 32, 58)( 33, 40)( 34, 39)( 35, 38)
( 36, 37)( 41,119)( 42,120)( 43,117)( 44,118)( 45,115)( 46,116)( 47,113)
( 48,114)( 49, 80)( 50, 79)( 51, 78)( 52, 77)( 53, 76)( 54, 75)( 55, 74)
( 56, 73)( 65,103)( 66,104)( 67,101)( 68,102)( 69, 99)( 70,100)( 71, 97)
( 72, 98)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(121,127)(122,128)(123,125)
(124,126);
poly := sub<Sym(128)|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*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0,
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2 >;

References : None.
to this polytope