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# Polytope of Type {108,6}

Atlas Canonical Name : {108,6}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1780)
Rank : 3
Schlafli Type : {108,6}
Number of vertices, edges, etc : 108, 324, 6
Order of s0s1s2 : 27
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {36,6}*432c
9-fold quotients : {12,6}*144d
27-fold quotients : {4,6}*48b
54-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 35)( 14, 36)
( 15, 33)( 16, 34)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 27)( 22, 28)
( 23, 25)( 24, 26)( 37,107)( 38,108)( 39,105)( 40,106)( 41,103)( 42,104)
( 43,101)( 44,102)( 45, 99)( 46,100)( 47, 97)( 48, 98)( 49, 95)( 50, 96)
( 51, 93)( 52, 94)( 53, 91)( 54, 92)( 55, 89)( 56, 90)( 57, 87)( 58, 88)
( 59, 85)( 60, 86)( 61, 83)( 62, 84)( 63, 81)( 64, 82)( 65, 79)( 66, 80)
( 67, 77)( 68, 78)( 69, 75)( 70, 76)( 71, 73)( 72, 74);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 45)(  6, 46)(  7, 48)(  8, 47)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 69)( 14, 70)( 15, 72)( 16, 71)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 61)( 22, 62)( 23, 64)( 24, 63)
( 25, 57)( 26, 58)( 27, 60)( 28, 59)( 29, 53)( 30, 54)( 31, 56)( 32, 55)
( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 73,105)( 74,106)( 75,108)( 76,107)
( 77,101)( 78,102)( 79,104)( 80,103)( 81, 97)( 82, 98)( 83,100)( 84, 99)
( 85, 93)( 86, 94)( 87, 96)( 88, 95)( 91, 92);;
s2 := (  2,  4)(  6,  8)( 10, 12)( 14, 16)( 18, 20)( 22, 24)( 26, 28)( 30, 32)
( 34, 36)( 38, 40)( 42, 44)( 46, 48)( 50, 52)( 54, 56)( 58, 60)( 62, 64)
( 66, 68)( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 86, 88)( 90, 92)( 94, 96)
( 98,100)(102,104)(106,108);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s2*s1*s0*s2*s1*s0*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 ];;
poly := F / rels;;

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

References : None.
to this polytope