Polytope of Type {6,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8}*96
Also Known As : {6,8|2}. if this polytope has another name.
Group : SmallGroup(96,117)
Rank : 3
Schlafli Type : {6,8}
Number of vertices, edges, etc : 6, 24, 8
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,8,2} of size 192
   {6,8,4} of size 384
   {6,8,4} of size 384
   {6,8,6} of size 576
   {6,8,3} of size 576
   {6,8,4} of size 768
   {6,8,8} of size 768
   {6,8,8} of size 768
   {6,8,8} of size 768
   {6,8,8} of size 768
   {6,8,4} of size 768
   {6,8,10} of size 960
   {6,8,12} of size 1152
   {6,8,12} of size 1152
   {6,8,3} of size 1152
   {6,8,6} of size 1152
   {6,8,6} of size 1152
   {6,8,14} of size 1344
   {6,8,18} of size 1728
   {6,8,9} of size 1728
   {6,8,6} of size 1728
   {6,8,20} of size 1920
   {6,8,20} of size 1920
Vertex Figure Of :
   {2,6,8} of size 192
   {3,6,8} of size 288
   {4,6,8} of size 384
   {3,6,8} of size 384
   {4,6,8} of size 384
   {6,6,8} of size 576
   {6,6,8} of size 576
   {6,6,8} of size 576
   {8,6,8} of size 768
   {4,6,8} of size 768
   {6,6,8} of size 768
   {9,6,8} of size 864
   {3,6,8} of size 864
   {5,6,8} of size 960
   {5,6,8} of size 960
   {10,6,8} of size 960
   {12,6,8} of size 1152
   {12,6,8} of size 1152
   {12,6,8} of size 1152
   {4,6,8} of size 1152
   {3,6,8} of size 1152
   {14,6,8} of size 1344
   {15,6,8} of size 1440
   {18,6,8} of size 1728
   {6,6,8} of size 1728
   {6,6,8} of size 1728
   {18,6,8} of size 1728
   {6,6,8} of size 1728
   {6,6,8} of size 1728
   {20,6,8} of size 1920
   {15,6,8} of size 1920
   {5,6,8} of size 1920
   {10,6,8} of size 1920
   {10,6,8} of size 1920
   {4,6,8} of size 1920
   {6,6,8} of size 1920
   {5,6,8} of size 1920
   {10,6,8} of size 1920
   {10,6,8} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*48a
   3-fold quotients : {2,8}*32
   4-fold quotients : {6,2}*24
   6-fold quotients : {2,4}*16
   8-fold quotients : {3,2}*12
   12-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,8}*192a, {6,16}*192
   3-fold covers : {18,8}*288, {6,24}*288a, {6,24}*288c
   4-fold covers : {24,8}*384b, {12,8}*384a, {24,8}*384d, {12,16}*384a, {12,16}*384b, {6,32}*384, {6,8}*384g
   5-fold covers : {6,40}*480, {30,8}*480
   6-fold covers : {36,8}*576a, {18,16}*576, {6,48}*576a, {12,24}*576b, {12,24}*576c, {6,48}*576c
   7-fold covers : {6,56}*672, {42,8}*672
   8-fold covers : {24,8}*768a, {12,8}*768a, {24,8}*768c, {12,16}*768a, {12,16}*768b, {48,8}*768a, {48,8}*768b, {24,16}*768c, {48,8}*768d, {24,16}*768d, {24,16}*768e, {48,8}*768f, {24,16}*768f, {12,32}*768a, {12,32}*768b, {6,64}*768, {6,8}*768j, {12,8}*768o, {12,8}*768u, {6,16}*768b, {6,16}*768c
   9-fold covers : {54,8}*864, {6,72}*864a, {18,24}*864a, {6,24}*864b, {18,24}*864b, {6,24}*864c, {6,24}*864f, {6,8}*864b
   10-fold covers : {6,80}*960, {12,40}*960a, {60,8}*960a, {30,16}*960
   11-fold covers : {6,88}*1056, {66,8}*1056
   12-fold covers : {36,8}*1152a, {12,24}*1152b, {12,24}*1152c, {72,8}*1152a, {72,8}*1152c, {24,24}*1152b, {24,24}*1152d, {24,24}*1152e, {24,24}*1152i, {36,16}*1152a, {12,48}*1152b, {12,48}*1152c, {36,16}*1152b, {12,48}*1152e, {12,48}*1152f, {18,32}*1152, {6,96}*1152a, {6,96}*1152c, {18,8}*1152g, {12,24}*1152o, {6,24}*1152h, {6,24}*1152j, {6,24}*1152k
   13-fold covers : {6,104}*1248, {78,8}*1248
   14-fold covers : {6,112}*1344, {12,56}*1344a, {84,8}*1344a, {42,16}*1344
   15-fold covers : {18,40}*1440, {90,8}*1440, {6,120}*1440a, {30,24}*1440b, {6,120}*1440b, {30,24}*1440c
   17-fold covers : {6,136}*1632, {102,8}*1632
   18-fold covers : {108,8}*1728a, {54,16}*1728, {6,144}*1728a, {18,48}*1728a, {6,48}*1728b, {36,24}*1728b, {12,24}*1728b, {12,72}*1728a, {36,24}*1728c, {12,24}*1728d, {18,48}*1728b, {6,48}*1728c, {6,48}*1728f, {12,24}*1728o, {12,8}*1728e, {6,16}*1728b, {12,8}*1728g, {12,24}*1728v
   19-fold covers : {6,152}*1824, {114,8}*1824
   20-fold covers : {60,8}*1920a, {12,40}*1920a, {120,8}*1920a, {120,8}*1920c, {24,40}*1920a, {24,40}*1920c, {60,16}*1920a, {12,80}*1920a, {60,16}*1920b, {12,80}*1920b, {30,32}*1920, {6,160}*1920, {6,40}*1920d, {30,8}*1920g
Permutation Representation (GAP) :

s0 := ( 3, 4)( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(23,24);;
s1 := ( 1, 3)( 2, 9)( 5, 6)( 7,10)( 8,15)(11,12)(13,16)(14,21)(17,18)(19,22)
(20,23);;
s2 := ( 1, 2)( 3, 6)( 4, 7)( 5, 8)( 9,12)(10,13)(11,14)(15,18)(16,19)(17,20)
(21,23)(22,24);;
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*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(24)!( 3, 4)( 6, 7)( 9,10)(12,13)(15,16)(18,19)(21,22)(23,24);
s1 := Sym(24)!( 1, 3)( 2, 9)( 5, 6)( 7,10)( 8,15)(11,12)(13,16)(14,21)(17,18)
(19,22)(20,23);
s2 := Sym(24)!( 1, 2)( 3, 6)( 4, 7)( 5, 8)( 9,12)(10,13)(11,14)(15,18)(16,19)
(17,20)(21,23)(22,24);
poly := sub<Sym(24)|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*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope