Polytope of Type {21,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6}*1344
if this polytope has a name.
Group : SmallGroup(1344,6320)
Rank : 3
Schlafli Type : {21,6}
Number of vertices, edges, etc : 112, 336, 32
Order of s₀s₁s₂ : 56
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {21,6}*336
   7-fold quotients : {3,6}*192
   28-fold quotients : {3,6}*48
   48-fold quotients : {7,2}*28
   56-fold quotients : {3,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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