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# Polytope of Type {2,3,6}

Atlas Canonical Name : {2,3,6}*72
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 4
Schlafli Type : {2,3,6}
Number of vertices, edges, etc : 2, 3, 9, 6
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{2,3,6,2} of size 144
{2,3,6,3} of size 216
{2,3,6,4} of size 288
{2,3,6,6} of size 432
{2,3,6,6} of size 432
{2,3,6,8} of size 576
{2,3,6,9} of size 648
{2,3,6,3} of size 648
{2,3,6,10} of size 720
{2,3,6,12} of size 864
{2,3,6,12} of size 864
{2,3,6,4} of size 864
{2,3,6,14} of size 1008
{2,3,6,15} of size 1080
{2,3,6,16} of size 1152
{2,3,6,4} of size 1152
{2,3,6,18} of size 1296
{2,3,6,6} of size 1296
{2,3,6,18} of size 1296
{2,3,6,6} of size 1296
{2,3,6,6} of size 1296
{2,3,6,20} of size 1440
{2,3,6,21} of size 1512
{2,3,6,22} of size 1584
{2,3,6,24} of size 1728
{2,3,6,24} of size 1728
{2,3,6,8} of size 1728
{2,3,6,26} of size 1872
{2,3,6,27} of size 1944
{2,3,6,9} of size 1944
Vertex Figure Of :
{2,2,3,6} of size 144
{3,2,3,6} of size 216
{4,2,3,6} of size 288
{5,2,3,6} of size 360
{6,2,3,6} of size 432
{7,2,3,6} of size 504
{8,2,3,6} of size 576
{9,2,3,6} of size 648
{10,2,3,6} of size 720
{11,2,3,6} of size 792
{12,2,3,6} of size 864
{13,2,3,6} of size 936
{14,2,3,6} of size 1008
{15,2,3,6} of size 1080
{16,2,3,6} of size 1152
{17,2,3,6} of size 1224
{18,2,3,6} of size 1296
{19,2,3,6} of size 1368
{20,2,3,6} of size 1440
{21,2,3,6} of size 1512
{22,2,3,6} of size 1584
{23,2,3,6} of size 1656
{24,2,3,6} of size 1728
{25,2,3,6} of size 1800
{26,2,3,6} of size 1872
{27,2,3,6} of size 1944
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {2,6,6}*144c
3-fold covers : {2,9,6}*216, {2,3,6}*216, {6,3,6}*216
4-fold covers : {2,12,6}*288b, {4,6,6}*288b, {2,6,12}*288c, {4,3,6}*288, {2,3,6}*288, {2,3,12}*288
5-fold covers : {2,15,6}*360
6-fold covers : {2,18,6}*432b, {2,6,6}*432c, {6,6,6}*432e, {6,6,6}*432f, {2,6,6}*432d
7-fold covers : {2,21,6}*504
8-fold covers : {4,12,6}*576b, {2,24,6}*576b, {8,6,6}*576b, {2,12,12}*576c, {2,6,24}*576c, {4,6,12}*576c, {2,3,12}*576, {2,3,24}*576, {8,3,6}*576, {4,6,6}*576b, {2,6,6}*576b, {2,6,12}*576b
9-fold covers : {2,9,18}*648, {2,9,6}*648a, {2,27,6}*648, {2,9,6}*648b, {2,9,6}*648c, {2,9,6}*648d, {2,3,6}*648, {2,3,18}*648, {6,9,6}*648, {6,3,6}*648a, {6,3,6}*648b
10-fold covers : {10,6,6}*720c, {2,6,30}*720a, {2,30,6}*720c
11-fold covers : {2,33,6}*792
12-fold covers : {2,36,6}*864b, {2,12,6}*864a, {4,18,6}*864b, {4,6,6}*864a, {2,18,12}*864b, {2,6,12}*864c, {2,9,6}*864, {4,9,6}*864, {2,9,12}*864, {2,3,6}*864, {2,3,12}*864, {4,3,6}*864, {6,12,6}*864d, {6,12,6}*864e, {12,6,6}*864c, {2,6,12}*864g, {2,12,6}*864g, {4,6,6}*864h, {6,6,12}*864f, {6,6,12}*864g, {12,6,6}*864g, {6,3,6}*864a, {6,3,6}*864b, {6,3,12}*864, {12,3,6}*864
13-fold covers : {2,39,6}*936
14-fold covers : {2,6,42}*1008a, {14,6,6}*1008c, {2,42,6}*1008c
15-fold covers : {2,45,6}*1080, {2,15,6}*1080, {6,15,6}*1080
16-fold covers : {4,12,12}*1152a, {8,12,6}*1152a, {4,24,6}*1152b, {2,24,12}*1152b, {2,12,24}*1152c, {8,12,6}*1152d, {4,24,6}*1152e, {2,24,12}*1152e, {2,12,24}*1152f, {4,12,6}*1152a, {2,12,12}*1152c, {8,6,12}*1152a, {4,6,24}*1152a, {16,6,6}*1152b, {2,6,48}*1152a, {2,48,6}*1152c, {2,3,6}*1152, {2,3,24}*1152, {4,3,6}*1152a, {8,3,6}*1152, {4,12,6}*1152f, {2,12,12}*1152e, {2,12,6}*1152a, {2,12,12}*1152h, {4,6,6}*1152c, {4,6,6}*1152e, {4,6,12}*1152c, {4,12,6}*1152i, {2,6,12}*1152c, {2,6,24}*1152b, {2,6,6}*1152b, {2,6,24}*1152d, {8,6,6}*1152c, {2,12,6}*1152d, {8,6,6}*1152e, {4,6,12}*1152d, {2,6,12}*1152e, {2,6,12}*1152f, {4,3,6}*1152b, {2,3,12}*1152, {2,6,6}*1152e, {4,3,12}*1152b
17-fold covers : {2,51,6}*1224
18-fold covers : {2,18,18}*1296c, {2,18,6}*1296a, {2,54,6}*1296b, {2,18,6}*1296c, {2,18,6}*1296d, {2,18,6}*1296e, {2,6,6}*1296d, {2,6,18}*1296h, {6,18,6}*1296c, {6,18,6}*1296d, {18,6,6}*1296d, {2,6,18}*1296i, {2,18,6}*1296i, {6,6,6}*1296d, {6,6,6}*1296e, {6,6,6}*1296l, {6,6,6}*1296m, {2,6,6}*1296e, {2,6,6}*1296f, {2,6,6}*1296g, {6,6,6}*1296q, {6,6,6}*1296r, {6,6,6}*1296t
19-fold covers : {2,57,6}*1368
20-fold covers : {10,12,6}*1440b, {20,6,6}*1440b, {2,6,60}*1440a, {2,12,30}*1440a, {10,6,12}*1440c, {4,6,30}*1440a, {2,60,6}*1440c, {4,30,6}*1440c, {2,30,12}*1440c, {4,15,6}*1440b, {2,15,12}*1440, {2,15,6}*1440e
21-fold covers : {2,63,6}*1512, {2,21,6}*1512, {6,21,6}*1512
22-fold covers : {2,6,66}*1584a, {22,6,6}*1584c, {2,66,6}*1584c
23-fold covers : {2,69,6}*1656
24-fold covers : {4,36,6}*1728b, {4,12,6}*1728a, {2,72,6}*1728b, {2,24,6}*1728a, {8,18,6}*1728b, {8,6,6}*1728a, {2,36,12}*1728b, {2,12,12}*1728a, {2,18,24}*1728b, {4,18,12}*1728b, {2,6,24}*1728c, {4,6,12}*1728c, {2,9,12}*1728, {2,9,24}*1728, {2,3,12}*1728, {2,3,24}*1728, {8,9,6}*1728, {8,3,6}*1728, {6,24,6}*1728d, {6,24,6}*1728e, {24,6,6}*1728c, {2,6,24}*1728f, {2,24,6}*1728f, {12,6,12}*1728d, {6,12,12}*1728e, {6,12,12}*1728f, {12,12,6}*1728d, {12,12,6}*1728e, {6,6,24}*1728f, {6,6,24}*1728g, {8,6,6}*1728e, {24,6,6}*1728g, {2,12,12}*1728h, {12,6,12}*1728g, {4,12,6}*1728j, {4,6,12}*1728h, {2,18,6}*1728, {4,18,6}*1728b, {2,18,12}*1728b, {4,6,6}*1728a, {2,6,6}*1728a, {2,6,12}*1728a, {6,3,12}*1728, {6,3,24}*1728, {12,3,6}*1728, {24,3,6}*1728, {4,6,6}*1728c, {6,6,6}*1728b, {6,6,6}*1728c, {6,6,6}*1728e, {6,6,12}*1728c, {6,6,12}*1728d, {2,6,6}*1728c, {6,12,6}*1728k, {2,6,12}*1728c, {12,6,6}*1728c, {12,6,6}*1728d, {2,12,6}*1728c
25-fold covers : {2,75,6}*1800, {10,3,6}*1800, {2,3,6}*1800, {2,3,30}*1800, {10,15,6}*1800, {2,15,30}*1800
26-fold covers : {2,6,78}*1872a, {26,6,6}*1872c, {2,78,6}*1872c
27-fold covers : {2,9,18}*1944a, {2,9,6}*1944a, {2,3,18}*1944a, {2,9,6}*1944b, {2,9,18}*1944b, {2,9,6}*1944c, {2,9,18}*1944c, {2,9,18}*1944d, {2,9,18}*1944e, {2,27,18}*1944, {2,27,6}*1944a, {2,9,6}*1944d, {2,9,18}*1944f, {2,9,18}*1944g, {2,9,18}*1944h, {2,9,18}*1944i, {2,9,6}*1944e, {2,9,18}*1944j, {2,27,6}*1944b, {2,27,6}*1944c, {2,81,6}*1944, {2,3,6}*1944, {2,3,18}*1944b, {6,9,18}*1944, {18,9,6}*1944, {6,9,6}*1944a, {6,9,6}*1944b, {6,3,6}*1944a, {6,3,6}*1944b, {6,3,6}*1944c, {6,27,6}*1944, {6,9,6}*1944c, {6,9,6}*1944d, {6,9,6}*1944e, {6,9,6}*1944f, {6,9,6}*1944g, {6,9,6}*1944h, {6,3,6}*1944d, {6,3,6}*1944e, {6,3,18}*1944, {18,3,6}*1944
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 7)( 8,11)( 9,10);;
s2 := ( 3, 8)( 4, 6)( 5,10)( 7, 9);;
s3 := ( 6, 7)( 8, 9)(10,11);;
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,
s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(11)!(1,2);
s1 := Sym(11)!( 4, 5)( 6, 7)( 8,11)( 9,10);
s2 := Sym(11)!( 3, 8)( 4, 6)( 5,10)( 7, 9);
s3 := Sym(11)!( 6, 7)( 8, 9)(10,11);
poly := sub<Sym(11)|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, s1*s2*s1*s2*s1*s2, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2 >;

to this polytope