Polytope of Type {3,6,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,2}*72
if this polytope has a name.
Group : SmallGroup(72,46)
Rank : 4
Schlafli Type : {3,6,2}
Number of vertices, edges, etc : 3, 9, 6, 2
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,2,2} of size 144
   {3,6,2,3} of size 216
   {3,6,2,4} of size 288
   {3,6,2,5} of size 360
   {3,6,2,6} of size 432
   {3,6,2,7} of size 504
   {3,6,2,8} of size 576
   {3,6,2,9} of size 648
   {3,6,2,10} of size 720
   {3,6,2,11} of size 792
   {3,6,2,12} of size 864
   {3,6,2,13} of size 936
   {3,6,2,14} of size 1008
   {3,6,2,15} of size 1080
   {3,6,2,16} of size 1152
   {3,6,2,17} of size 1224
   {3,6,2,18} of size 1296
   {3,6,2,19} of size 1368
   {3,6,2,20} of size 1440
   {3,6,2,21} of size 1512
   {3,6,2,22} of size 1584
   {3,6,2,23} of size 1656
   {3,6,2,24} of size 1728
   {3,6,2,25} of size 1800
   {3,6,2,26} of size 1872
   {3,6,2,27} of size 1944
Vertex Figure Of :
   {2,3,6,2} of size 144
   {4,3,6,2} of size 288
   {6,3,6,2} of size 432
   {4,3,6,2} of size 576
   {8,3,6,2} of size 1152
   {6,3,6,2} of size 1296
   {6,3,6,2} of size 1728
   {12,3,6,2} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,4}*144, {6,6,2}*144c
   3-fold covers : {9,6,2}*216, {3,6,2}*216, {3,6,6}*216b
   4-fold covers : {3,6,8}*288, {12,6,2}*288b, {6,6,4}*288c, {6,12,2}*288c, {3,6,2}*288, {3,12,2}*288
   5-fold covers : {3,6,10}*360, {15,6,2}*360
   6-fold covers : {9,6,4}*432, {3,6,4}*432a, {18,6,2}*432b, {6,6,2}*432c, {3,6,12}*432b, {6,6,6}*432g, {6,6,2}*432d
   7-fold covers : {3,6,14}*504, {21,6,2}*504
   8-fold covers : {3,6,16}*576, {24,6,2}*576b, {12,12,2}*576c, {12,6,4}*576b, {6,6,8}*576c, {6,24,2}*576c, {6,12,4}*576c, {3,12,2}*576, {3,24,2}*576, {3,6,4}*576a, {3,12,4}*576, {6,6,2}*576b, {6,12,2}*576b
   9-fold covers : {9,18,2}*648, {9,6,2}*648a, {27,6,2}*648, {9,6,2}*648b, {9,6,2}*648c, {9,6,2}*648d, {3,6,2}*648, {3,18,2}*648, {3,6,18}*648b, {9,6,6}*648b, {3,6,6}*648c, {3,6,6}*648d, {3,6,6}*648e
   10-fold covers : {3,6,20}*720, {15,6,4}*720, {6,6,10}*720c, {6,30,2}*720a, {30,6,2}*720c
   11-fold covers : {3,6,22}*792, {33,6,2}*792
   12-fold covers : {9,6,8}*864, {3,6,8}*864a, {36,6,2}*864b, {12,6,2}*864a, {18,6,4}*864b, {18,12,2}*864b, {6,6,4}*864c, {6,12,2}*864c, {3,6,24}*864b, {9,6,2}*864, {9,12,2}*864, {3,6,2}*864, {3,12,2}*864, {12,6,6}*864d, {6,6,12}*864e, {6,12,2}*864g, {12,6,2}*864g, {6,6,4}*864h, {6,12,6}*864f, {3,6,6}*864, {3,12,6}*864b
   13-fold covers : {3,6,26}*936, {39,6,2}*936
   14-fold covers : {3,6,28}*1008, {21,6,4}*1008, {6,6,14}*1008c, {6,42,2}*1008a, {42,6,2}*1008c
   15-fold covers : {9,6,10}*1080, {3,6,10}*1080, {45,6,2}*1080, {15,6,2}*1080, {3,6,30}*1080b, {15,6,6}*1080b
   16-fold covers : {3,6,32}*1152, {12,12,4}*1152c, {6,12,8}*1152c, {6,24,4}*1152a, {24,12,2}*1152b, {12,24,2}*1152c, {6,12,8}*1152f, {6,24,4}*1152d, {24,12,2}*1152e, {12,24,2}*1152f, {6,12,4}*1152c, {12,12,2}*1152c, {12,6,8}*1152c, {24,6,4}*1152c, {6,6,16}*1152c, {6,48,2}*1152a, {48,6,2}*1152c, {3,6,2}*1152, {3,24,2}*1152, {3,6,4}*1152a, {3,12,4}*1152a, {3,6,8}*1152, {3,12,8}*1152, {3,12,4}*1152b, {3,24,4}*1152, {12,12,2}*1152e, {12,6,2}*1152a, {12,12,2}*1152h, {6,12,2}*1152c, {6,24,2}*1152b, {6,6,2}*1152b, {6,24,2}*1152d, {12,6,2}*1152d, {6,6,4}*1152f, {6,12,4}*1152j, {6,12,2}*1152e, {6,12,2}*1152f, {3,12,2}*1152, {3,6,4}*1152c, {6,6,2}*1152e
   17-fold covers : {3,6,34}*1224, {51,6,2}*1224
   18-fold covers : {9,18,4}*1296, {9,6,4}*1296a, {27,6,4}*1296, {9,6,4}*1296b, {9,6,4}*1296c, {9,6,4}*1296d, {3,6,4}*1296a, {3,18,4}*1296, {18,18,2}*1296c, {18,6,2}*1296a, {54,6,2}*1296b, {18,6,2}*1296c, {18,6,2}*1296d, {18,6,2}*1296e, {6,6,2}*1296d, {6,18,2}*1296h, {3,6,36}*1296b, {9,6,12}*1296b, {3,6,12}*1296c, {3,6,12}*1296d, {3,6,12}*1296e, {6,6,18}*1296e, {18,6,6}*1296e, {6,18,2}*1296i, {18,6,2}*1296i, {6,6,6}*1296c, {6,6,6}*1296o, {6,6,6}*1296p, {6,6,2}*1296e, {6,6,2}*1296f, {6,6,2}*1296g, {3,6,4}*1296b, {6,6,6}*1296s, {6,6,6}*1296t
   19-fold covers : {3,6,38}*1368, {57,6,2}*1368
   20-fold covers : {3,6,40}*1440, {15,6,8}*1440, {12,6,10}*1440b, {6,6,20}*1440c, {6,60,2}*1440a, {12,30,2}*1440a, {6,12,10}*1440c, {6,30,4}*1440a, {60,6,2}*1440c, {30,6,4}*1440c, {30,12,2}*1440c, {3,6,10}*1440, {3,12,10}*1440, {15,12,2}*1440, {15,6,2}*1440e
   21-fold covers : {9,6,14}*1512, {3,6,14}*1512, {63,6,2}*1512, {21,6,2}*1512, {3,6,42}*1512b, {21,6,6}*1512b
   22-fold covers : {3,6,44}*1584, {33,6,4}*1584, {6,6,22}*1584c, {6,66,2}*1584a, {66,6,2}*1584c
   23-fold covers : {3,6,46}*1656, {69,6,2}*1656
   24-fold covers : {9,6,16}*1728, {3,6,16}*1728a, {72,6,2}*1728b, {24,6,2}*1728a, {36,12,2}*1728b, {36,6,4}*1728b, {12,12,2}*1728a, {12,6,4}*1728b, {18,6,8}*1728b, {18,24,2}*1728b, {6,6,8}*1728c, {6,24,2}*1728c, {18,12,4}*1728b, {6,12,4}*1728c, {3,6,48}*1728b, {9,12,2}*1728, {9,6,4}*1728a, {9,24,2}*1728, {3,12,2}*1728, {3,24,2}*1728, {9,12,4}*1728, {3,6,4}*1728a, {3,12,4}*1728a, {24,6,6}*1728d, {6,6,24}*1728e, {6,24,2}*1728f, {24,6,2}*1728f, {12,6,12}*1728f, {12,12,6}*1728f, {6,6,8}*1728e, {6,24,6}*1728g, {12,12,2}*1728h, {6,12,4}*1728j, {6,12,12}*1728g, {12,6,4}*1728h, {18,6,2}*1728, {18,12,2}*1728b, {6,6,2}*1728a, {6,12,2}*1728a, {3,12,6}*1728, {3,24,6}*1728b, {3,6,12}*1728, {3,12,12}*1728b, {6,6,4}*1728c, {6,6,6}*1728f, {6,12,6}*1728h, {6,12,6}*1728l, {6,6,2}*1728c, {6,12,2}*1728c, {12,6,2}*1728c
   25-fold covers : {3,6,50}*1800, {75,6,2}*1800, {3,6,2}*1800, {3,30,2}*1800, {15,6,10}*1800, {15,30,2}*1800
   26-fold covers : {3,6,52}*1872, {39,6,4}*1872, {6,6,26}*1872c, {6,78,2}*1872a, {78,6,2}*1872c
   27-fold covers : {9,18,2}*1944a, {9,6,2}*1944a, {3,18,2}*1944a, {9,6,2}*1944b, {9,18,2}*1944b, {9,6,2}*1944c, {9,18,2}*1944c, {9,18,2}*1944d, {9,18,2}*1944e, {27,18,2}*1944, {27,6,2}*1944a, {9,6,2}*1944d, {9,18,2}*1944f, {9,18,2}*1944g, {9,18,2}*1944h, {9,18,2}*1944i, {9,6,2}*1944e, {9,18,2}*1944j, {27,6,2}*1944b, {27,6,2}*1944c, {81,6,2}*1944, {3,6,2}*1944, {3,18,2}*1944b, {9,6,18}*1944b, {9,18,6}*1944, {3,6,18}*1944c, {3,6,18}*1944d, {9,6,6}*1944c, {9,6,6}*1944d, {3,6,18}*1944e, {9,6,6}*1944e, {3,6,6}*1944b, {3,6,6}*1944c, {3,6,6}*1944d, {3,6,54}*1944b, {27,6,6}*1944b, {3,6,6}*1944e, {3,6,6}*1944f, {3,6,6}*1944g, {9,6,6}*1944f, {9,6,6}*1944g, {9,6,6}*1944h, {3,6,6}*1944h, {3,18,6}*1944
Permutation Representation (GAP) :

s0 := (2,3)(4,5)(6,9)(7,8);;
s1 := (1,6)(2,4)(3,8)(5,7);;
s2 := (4,5)(6,7)(8,9);;
s3 := (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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(11)!(2,3)(4,5)(6,9)(7,8);
s1 := Sym(11)!(1,6)(2,4)(3,8)(5,7);
s2 := Sym(11)!(4,5)(6,7)(8,9);
s3 := Sym(11)!(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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >;

to this polytope