Polytope of Type {56,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {56,6}*1344a
if this polytope has a name.
Group : SmallGroup(1344,6320)
Rank : 3
Schlafli Type : {56,6}
Number of vertices, edges, etc : 112, 336, 12
Order of s₀s₁s₂ : 21
Order of s₀s₁s₂s₁ : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
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) :
   4-fold quotients : {28,6}*336b
   7-fold quotients : {8,6}*192a
   28-fold quotients : {4,6}*48b
   56-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      8 facets:
         4 of {28}*56
         4 of {56}*112
      56 vertex figures:
         56 of {6}*12

Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope

Polytope of Type {56,6}

Twisty Puzzle