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Polytope of Type {8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6}*96
Also Known As : {8,6|2}. if this polytope has another name.
Group : SmallGroup(96,117)
Rank : 3
Schlafli Type : {8,6}
Number of vertices, edges, etc : 8, 24, 6
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 :
   {8,6,2} of size 192
   {8,6,3} of size 288
   {8,6,4} of size 384
   {8,6,3} of size 384
   {8,6,4} of size 384
   {8,6,6} of size 576
   {8,6,6} of size 576
   {8,6,6} of size 576
   {8,6,8} of size 768
   {8,6,4} of size 768
   {8,6,6} of size 768
   {8,6,9} of size 864
   {8,6,3} of size 864
   {8,6,5} of size 960
   {8,6,5} of size 960
   {8,6,10} of size 960
   {8,6,12} of size 1152
   {8,6,12} of size 1152
   {8,6,12} of size 1152
   {8,6,4} of size 1152
   {8,6,3} of size 1152
   {8,6,14} of size 1344
   {8,6,15} of size 1440
   {8,6,18} of size 1728
   {8,6,6} of size 1728
   {8,6,6} of size 1728
   {8,6,18} of size 1728
   {8,6,6} of size 1728
   {8,6,6} of size 1728
   {8,6,20} of size 1920
   {8,6,15} of size 1920
   {8,6,5} of size 1920
   {8,6,10} of size 1920
   {8,6,10} of size 1920
   {8,6,4} of size 1920
   {8,6,6} of size 1920
   {8,6,5} of size 1920
   {8,6,10} of size 1920
   {8,6,10} of size 1920
Vertex Figure Of :
   {2,8,6} of size 192
   {4,8,6} of size 384
   {4,8,6} of size 384
   {6,8,6} of size 576
   {3,8,6} of size 576
   {4,8,6} of size 768
   {8,8,6} of size 768
   {8,8,6} of size 768
   {8,8,6} of size 768
   {8,8,6} of size 768
   {4,8,6} of size 768
   {10,8,6} of size 960
   {12,8,6} of size 1152
   {12,8,6} of size 1152
   {3,8,6} of size 1152
   {6,8,6} of size 1152
   {6,8,6} of size 1152
   {14,8,6} of size 1344
   {18,8,6} of size 1728
   {9,8,6} of size 1728
   {6,8,6} of size 1728
   {20,8,6} of size 1920
   {20,8,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*48a
   3-fold quotients : {8,2}*32
   4-fold quotients : {2,6}*24
   6-fold quotients : {4,2}*16
   8-fold quotients : {2,3}*12
   12-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,12}*192a, {16,6}*192
   3-fold covers : {8,18}*288, {24,6}*288a, {24,6}*288c
   4-fold covers : {8,24}*384b, {8,12}*384a, {8,24}*384d, {16,12}*384a, {16,12}*384b, {32,6}*384, {8,6}*384g
   5-fold covers : {40,6}*480, {8,30}*480
   6-fold covers : {8,36}*576a, {16,18}*576, {48,6}*576a, {24,12}*576b, {24,12}*576c, {48,6}*576c
   7-fold covers : {56,6}*672, {8,42}*672
   8-fold covers : {8,24}*768a, {8,12}*768a, {8,24}*768c, {16,12}*768a, {16,12}*768b, {8,48}*768a, {8,48}*768b, {16,24}*768c, {8,48}*768d, {16,24}*768d, {16,24}*768e, {8,48}*768f, {16,24}*768f, {32,12}*768a, {32,12}*768b, {64,6}*768, {8,6}*768j, {8,12}*768o, {8,12}*768u, {16,6}*768b, {16,6}*768c
   9-fold covers : {8,54}*864, {72,6}*864a, {24,18}*864a, {24,6}*864b, {24,18}*864b, {24,6}*864c, {24,6}*864f, {8,6}*864b
   10-fold covers : {80,6}*960, {40,12}*960a, {8,60}*960a, {16,30}*960
   11-fold covers : {88,6}*1056, {8,66}*1056
   12-fold covers : {8,36}*1152a, {24,12}*1152b, {24,12}*1152c, {8,72}*1152a, {8,72}*1152c, {24,24}*1152b, {24,24}*1152f, {24,24}*1152g, {24,24}*1152h, {16,36}*1152a, {48,12}*1152b, {48,12}*1152c, {16,36}*1152b, {48,12}*1152e, {48,12}*1152f, {32,18}*1152, {96,6}*1152a, {96,6}*1152c, {8,18}*1152g, {24,12}*1152o, {24,6}*1152h, {24,6}*1152j, {24,6}*1152k
   13-fold covers : {104,6}*1248, {8,78}*1248
   14-fold covers : {112,6}*1344, {56,12}*1344a, {8,84}*1344a, {16,42}*1344
   15-fold covers : {40,18}*1440, {8,90}*1440, {120,6}*1440a, {24,30}*1440b, {120,6}*1440b, {24,30}*1440c
   17-fold covers : {136,6}*1632, {8,102}*1632
   18-fold covers : {8,108}*1728a, {16,54}*1728, {144,6}*1728a, {48,18}*1728a, {48,6}*1728b, {24,36}*1728b, {24,12}*1728b, {72,12}*1728a, {24,36}*1728c, {24,12}*1728d, {48,18}*1728b, {48,6}*1728c, {48,6}*1728f, {24,12}*1728o, {8,12}*1728e, {16,6}*1728b, {8,12}*1728g, {24,12}*1728v
   19-fold covers : {152,6}*1824, {8,114}*1824
   20-fold covers : {8,60}*1920a, {40,12}*1920a, {8,120}*1920a, {8,120}*1920c, {40,24}*1920a, {40,24}*1920c, {16,60}*1920a, {80,12}*1920a, {16,60}*1920b, {80,12}*1920b, {32,30}*1920, {160,6}*1920, {40,6}*1920d, {8,30}*1920g
Permutation Representation (GAP) :
s0 := ( 2, 5)( 6, 9)( 7,10)( 8,11)(12,15)(13,16)(14,17)(18,21)(19,22);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 5, 8)( 9,13)(10,12)(11,14)(15,19)(16,18)(17,20)
(21,24)(22,23);;
s2 := ( 1, 3)( 2, 6)( 5, 9)( 8,12)(11,15)(14,18)(17,21)(20,23);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(24)!( 2, 5)( 6, 9)( 7,10)( 8,11)(12,15)(13,16)(14,17)(18,21)(19,22);
s1 := Sym(24)!( 1, 2)( 3, 7)( 4, 6)( 5, 8)( 9,13)(10,12)(11,14)(15,19)(16,18)
(17,20)(21,24)(22,23);
s2 := Sym(24)!( 1, 3)( 2, 6)( 5, 9)( 8,12)(11,15)(14,18)(17,21)(20,23);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >; 
 
References : None.
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