Polytope of Type {6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*144
Also Known As : {6,4}₄. if this polytope has another name.
Group : SmallGroup(144,186)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 18, 36, 12
Order of s₀s₁s₂ : 4
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   {6,4,2} of size 288
   {6,4,4} of size 576
   {6,4,6} of size 864
   {6,4,8} of size 1152
   {6,4,8} of size 1152
   {6,4,4} of size 1152
   {6,4,3} of size 1296
   {6,4,4} of size 1296
   {6,4,10} of size 1440
   {6,4,12} of size 1728
Vertex Figure Of :
   {2,6,4} of size 288
   {3,6,4} of size 432
   {4,6,4} of size 576
   {6,6,4} of size 864
   {6,6,4} of size 864
   {8,6,4} of size 1152
   {9,6,4} of size 1296
   {10,6,4} of size 1440
   {12,6,4} of size 1728
   {12,6,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*72
   9-fold quotients : {2,4}*16
   18-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,8}*288, {12,4}*288
   3-fold covers : {6,4}*432a, {6,12}*432e, {6,12}*432f, {6,4}*432b, {6,12}*432h, {6,12}*432i
   4-fold covers : {6,16}*576, {12,4}*576, {12,8}*576a, {24,4}*576a, {24,4}*576b, {12,8}*576b
   5-fold covers : {30,4}*720, {6,20}*720
   6-fold covers : {6,8}*864a, {6,24}*864d, {6,24}*864e, {12,4}*864b, {12,12}*864f, {12,12}*864g, {12,4}*864d, {12,12}*864j, {6,8}*864b, {6,24}*864g, {6,24}*864h, {12,12}*864l
   7-fold covers : {42,4}*1008, {6,28}*1008
   8-fold covers : {24,4}*1152a, {12,8}*1152a, {24,8}*1152a, {24,8}*1152b, {24,8}*1152c, {24,8}*1152d, {48,4}*1152a, {12,16}*1152a, {48,4}*1152b, {12,16}*1152b, {12,4}*1152a, {12,8}*1152b, {24,4}*1152b, {6,32}*1152
   9-fold covers : {18,4}*1296a, {18,4}*1296b, {6,4}*1296a, {6,12}*1296j, {6,12}*1296k, {6,12}*1296l, {6,12}*1296m, {6,12}*1296n, {6,36}*1296m, {6,12}*1296o, {6,36}*1296n, {6,36}*1296o, {6,12}*1296s, {6,12}*1296t, {6,12}*1296u
   10-fold covers : {60,4}*1440, {30,8}*1440, {6,40}*1440, {12,20}*1440
   11-fold covers : {66,4}*1584, {6,44}*1584
   12-fold covers : {6,16}*1728a, {6,48}*1728d, {6,48}*1728e, {12,4}*1728b, {12,12}*1728f, {12,12}*1728g, {12,8}*1728a, {12,24}*1728g, {12,24}*1728h, {24,4}*1728a, {24,12}*1728i, {24,12}*1728j, {24,4}*1728c, {24,12}*1728k, {24,12}*1728l, {12,8}*1728d, {12,24}*1728m, {12,24}*1728n, {24,4}*1728f, {24,12}*1728q, {24,4}*1728g, {24,12}*1728r, {6,16}*1728b, {6,48}*1728g, {12,8}*1728g, {12,24}*1728s, {12,8}*1728h, {12,24}*1728t, {12,4}*1728c, {12,12}*1728q, {6,48}*1728h, {12,12}*1728t, {12,24}*1728u, {24,12}*1728v, {24,12}*1728w, {12,24}*1728x, {6,4}*1728, {6,12}*1728j, {12,12}*1728ab
   13-fold covers : {78,4}*1872, {6,52}*1872
Permutation Representation (GAP) :

s0 := ( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)(10,19)
(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(37,64)(38,66)(39,65)
(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,55)(47,57)(48,56)(49,61)(50,63)
(51,62)(52,58)(53,60)(54,59);;
s1 := ( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)
(29,32)(30,33)(37,49)(38,50)(39,51)(40,46)(41,47)(42,48)(43,52)(44,53)(45,54)
(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(61,70)(62,71)(63,72);;
s2 := ( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)(10,46)
(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)(21,61)
(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)(32,68)
(33,71)(34,66)(35,69)(36,72);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(72)!( 1,28)( 2,30)( 3,29)( 4,34)( 5,36)( 6,35)( 7,31)( 8,33)( 9,32)
(10,19)(11,21)(12,20)(13,25)(14,27)(15,26)(16,22)(17,24)(18,23)(37,64)(38,66)
(39,65)(40,70)(41,72)(42,71)(43,67)(44,69)(45,68)(46,55)(47,57)(48,56)(49,61)
(50,63)(51,62)(52,58)(53,60)(54,59);
s1 := Sym(72)!( 1, 4)( 2, 5)( 3, 6)(10,13)(11,14)(12,15)(19,22)(20,23)(21,24)
(28,31)(29,32)(30,33)(37,49)(38,50)(39,51)(40,46)(41,47)(42,48)(43,52)(44,53)
(45,54)(55,67)(56,68)(57,69)(58,64)(59,65)(60,66)(61,70)(62,71)(63,72);
s2 := Sym(72)!( 1,37)( 2,40)( 3,43)( 4,38)( 5,41)( 6,44)( 7,39)( 8,42)( 9,45)
(10,46)(11,49)(12,52)(13,47)(14,50)(15,53)(16,48)(17,51)(18,54)(19,55)(20,58)
(21,61)(22,56)(23,59)(24,62)(25,57)(26,60)(27,63)(28,64)(29,67)(30,70)(31,65)
(32,68)(33,71)(34,66)(35,69)(36,72);
poly := sub<Sym(72)|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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope