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# Polytope of Type {2,4,3}

Atlas Canonical Name : {2,4,3}*96
if this polytope has a name.
Group : SmallGroup(96,226)
Rank : 4
Schlafli Type : {2,4,3}
Number of vertices, edges, etc : 2, 8, 12, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,4,3,2} of size 192
{2,4,3,3} of size 384
{2,4,3,4} of size 384
{2,4,3,6} of size 576
{2,4,3,3} of size 768
{2,4,3,6} of size 768
{2,4,3,6} of size 768
{2,4,3,4} of size 768
{2,4,3,6} of size 1728
Vertex Figure Of :
{2,2,4,3} of size 192
{3,2,4,3} of size 288
{4,2,4,3} of size 384
{5,2,4,3} of size 480
{6,2,4,3} of size 576
{7,2,4,3} of size 672
{8,2,4,3} of size 768
{9,2,4,3} of size 864
{10,2,4,3} of size 960
{11,2,4,3} of size 1056
{12,2,4,3} of size 1152
{13,2,4,3} of size 1248
{14,2,4,3} of size 1344
{15,2,4,3} of size 1440
{17,2,4,3} of size 1632
{18,2,4,3} of size 1728
{19,2,4,3} of size 1824
{20,2,4,3} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,3}*48
4-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,3}*192b, {2,8,3}*192, {2,4,6}*192
3-fold covers : {2,4,9}*288, {6,4,3}*288, {2,12,3}*288
4-fold covers : {4,4,3}*384b, {2,8,3}*384, {4,8,3}*384, {8,4,3}*384, {2,4,12}*384b, {4,4,6}*384d, {2,4,6}*384b, {2,4,12}*384c, {2,8,6}*384b, {2,8,6}*384c
5-fold covers : {10,4,3}*480, {2,4,15}*480
6-fold covers : {4,4,9}*576b, {2,8,9}*576, {2,4,18}*576, {12,4,3}*576, {2,24,3}*576, {6,8,3}*576, {4,12,3}*576, {6,4,6}*576a, {2,12,6}*576a, {2,12,6}*576b
7-fold covers : {14,4,3}*672, {2,4,21}*672
8-fold covers : {2,8,3}*768, {2,8,6}*768a, {4,8,3}*768a, {4,8,3}*768b, {8,8,3}*768, {4,4,3}*768a, {4,4,6}*768b, {4,8,3}*768c, {4,8,3}*768d, {4,4,3}*768b, {4,4,3}*768c, {4,8,3}*768e, {4,8,3}*768f, {16,4,3}*768, {2,4,12}*768d, {4,4,6}*768e, {4,4,12}*768e, {4,4,12}*768f, {2,8,6}*768d, {2,8,6}*768e, {4,4,6}*768f, {2,4,6}*768a, {2,8,12}*768e, {2,8,12}*768f, {2,4,24}*768c, {2,4,24}*768d, {4,8,6}*768c, {2,8,6}*768f, {2,8,12}*768g, {2,8,12}*768h, {8,4,6}*768c, {2,8,6}*768g, {4,8,6}*768d, {2,4,6}*768b, {2,4,24}*768e, {2,4,12}*768e, {2,4,24}*768f
9-fold covers : {2,4,27}*864, {18,4,3}*864, {6,4,9}*864, {2,12,9}*864, {2,12,3}*864, {6,12,3}*864a, {6,12,3}*864b
10-fold covers : {20,4,3}*960, {10,8,3}*960, {4,4,15}*960b, {2,8,15}*960, {10,4,6}*960, {2,20,6}*960c, {2,4,30}*960
11-fold covers : {22,4,3}*1056, {2,4,33}*1056
12-fold covers : {4,4,9}*1152b, {2,8,9}*1152, {4,8,9}*1152, {8,4,9}*1152, {2,4,36}*1152b, {4,4,18}*1152d, {2,4,18}*1152b, {2,4,36}*1152c, {2,8,18}*1152b, {2,8,18}*1152c, {2,24,3}*1152, {12,8,3}*1152, {4,12,3}*1152a, {12,4,3}*1152, {6,8,3}*1152, {24,4,3}*1152, {8,12,3}*1152, {4,24,3}*1152, {6,4,12}*1152b, {2,12,12}*1152f, {2,12,12}*1152g, {12,4,6}*1152c, {2,12,6}*1152b, {2,12,12}*1152i, {4,12,6}*1152g, {6,4,6}*1152a, {6,4,12}*1152d, {2,24,6}*1152b, {2,24,6}*1152c, {2,24,6}*1152d, {6,8,6}*1152a, {2,24,6}*1152e, {6,8,6}*1152c, {4,12,6}*1152j, {2,12,6}*1152f, {2,12,12}*1152k, {2,12,3}*1152
13-fold covers : {26,4,3}*1248, {2,4,39}*1248
14-fold covers : {28,4,3}*1344, {14,8,3}*1344, {4,4,21}*1344b, {2,8,21}*1344, {14,4,6}*1344, {2,28,6}*1344, {2,4,42}*1344
15-fold covers : {10,4,9}*1440, {2,4,45}*1440, {10,12,3}*1440, {6,4,15}*1440, {2,12,15}*1440, {30,4,3}*1440
17-fold covers : {34,4,3}*1632, {2,4,51}*1632
18-fold covers : {4,4,27}*1728b, {2,8,27}*1728, {2,4,54}*1728, {36,4,3}*1728, {18,8,3}*1728, {12,4,9}*1728, {12,12,3}*1728a, {2,24,9}*1728, {2,24,3}*1728, {6,8,9}*1728, {6,24,3}*1728a, {4,12,9}*1728, {4,12,3}*1728a, {18,4,6}*1728a, {2,36,6}*1728, {6,4,18}*1728b, {2,12,18}*1728a, {2,12,18}*1728b, {6,12,6}*1728a, {2,12,6}*1728a, {2,12,6}*1728b, {6,24,3}*1728b, {12,12,3}*1728b, {4,12,3}*1728b, {6,12,6}*1728e, {6,12,6}*1728f, {6,12,6}*1728g, {6,12,6}*1728h, {2,12,6}*1728c
19-fold covers : {38,4,3}*1824, {2,4,57}*1824
20-fold covers : {20,8,3}*1920, {20,4,3}*1920, {10,8,3}*1920, {40,4,3}*1920, {4,4,15}*1920b, {2,8,15}*1920a, {4,8,15}*1920, {8,4,15}*1920, {10,4,12}*1920b, {2,20,12}*1920b, {20,4,6}*1920b, {2,20,6}*1920a, {4,20,6}*1920c, {10,4,6}*1920, {10,4,12}*1920c, {2,40,6}*1920b, {10,8,6}*1920a, {2,40,6}*1920c, {10,8,6}*1920b, {2,20,12}*1920c, {2,4,60}*1920b, {4,4,30}*1920d, {2,4,30}*1920b, {2,4,60}*1920c, {2,8,30}*1920b, {2,8,30}*1920c
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (6,8);;
s2 := (5,6)(7,8);;
s3 := (3,5)(4,7);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(8)!(1,2);
s1 := Sym(8)!(6,8);
s2 := Sym(8)!(5,6)(7,8);
s3 := Sym(8)!(3,5)(4,7);
poly := sub<Sym(8)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope