Polytope of Type {10,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {10,6}*120
Also Known As : {10,6|2}. if this polytope has another name.
Group : SmallGroup(120,42)
Rank : 3
Schlafli Type : {10,6}
Number of vertices, edges, etc : 10, 30, 6
Order of s₀s₁s₂ : 30
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {10,6,2} of size 240
   {10,6,3} of size 360
   {10,6,4} of size 480
   {10,6,3} of size 480
   {10,6,4} of size 480
   {10,6,6} of size 720
   {10,6,6} of size 720
   {10,6,6} of size 720
   {10,6,8} of size 960
   {10,6,4} of size 960
   {10,6,6} of size 960
   {10,6,9} of size 1080
   {10,6,3} of size 1080
   {10,6,5} of size 1200
   {10,6,5} of size 1200
   {10,6,10} of size 1200
   {10,6,12} of size 1440
   {10,6,12} of size 1440
   {10,6,12} of size 1440
   {10,6,3} of size 1440
   {10,6,4} of size 1440
   {10,6,14} of size 1680
   {10,6,15} of size 1800
   {10,6,16} of size 1920
   {10,6,4} of size 1920
   {10,6,3} of size 1920
   {10,6,4} of size 1920
   {10,6,12} of size 1920
   {10,6,8} of size 1920
   {10,6,12} of size 1920
   {10,6,6} of size 1920
   {10,6,8} of size 1920
Vertex Figure Of :
   {2,10,6} of size 240
   {4,10,6} of size 480
   {5,10,6} of size 600
   {3,10,6} of size 720
   {5,10,6} of size 720
   {6,10,6} of size 720
   {8,10,6} of size 960
   {10,10,6} of size 1200
   {10,10,6} of size 1200
   {10,10,6} of size 1200
   {12,10,6} of size 1440
   {4,10,6} of size 1440
   {6,10,6} of size 1440
   {3,10,6} of size 1440
   {5,10,6} of size 1440
   {6,10,6} of size 1440
   {6,10,6} of size 1440
   {10,10,6} of size 1440
   {10,10,6} of size 1440
   {14,10,6} of size 1680
   {3,10,6} of size 1800
   {15,10,6} of size 1800
   {16,10,6} of size 1920
   {4,10,6} of size 1920
   {5,10,6} of size 1920
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {10,2}*40
   5-fold quotients : {2,6}*24
   6-fold quotients : {5,2}*20
   10-fold quotients : {2,3}*12
   15-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {10,12}*240, {20,6}*240a
   3-fold covers : {10,18}*360, {30,6}*360a, {30,6}*360b
   4-fold covers : {10,24}*480, {40,6}*480, {20,12}*480, {20,6}*480c
   5-fold covers : {50,6}*600, {10,30}*600a, {10,30}*600b
   6-fold covers : {10,36}*720, {20,18}*720a, {60,6}*720a, {30,12}*720a, {30,12}*720b, {60,6}*720b
   7-fold covers : {10,42}*840, {70,6}*840
   8-fold covers : {10,48}*960, {80,6}*960, {20,12}*960a, {20,24}*960a, {40,12}*960a, {20,24}*960b, {40,12}*960b, {20,12}*960b, {20,6}*960e, {40,6}*960d, {40,6}*960e, {20,12}*960c
   9-fold covers : {10,54}*1080, {30,18}*1080a, {30,6}*1080a, {90,6}*1080a, {30,18}*1080b, {30,6}*1080c, {30,6}*1080d
   10-fold covers : {50,12}*1200, {100,6}*1200a, {20,30}*1200a, {10,60}*1200a, {20,30}*1200b, {10,60}*1200b
   11-fold covers : {10,66}*1320, {110,6}*1320
   12-fold covers : {10,72}*1440, {40,18}*1440, {20,36}*1440, {120,6}*1440a, {30,24}*1440a, {60,12}*1440a, {30,24}*1440b, {120,6}*1440b, {60,12}*1440b, {20,18}*1440, {30,6}*1440g, {60,6}*1440c, {30,12}*1440a, {60,6}*1440d
   13-fold covers : {10,78}*1560, {130,6}*1560
   14-fold covers : {20,42}*1680a, {10,84}*1680, {70,12}*1680, {140,6}*1680a
   15-fold covers : {50,18}*1800, {150,6}*1800a, {150,6}*1800b, {10,90}*1800a, {10,90}*1800b, {30,30}*1800c, {30,30}*1800e, {30,30}*1800f, {30,30}*1800g
   16-fold covers : {40,12}*1920a, {20,24}*1920a, {40,24}*1920a, {40,24}*1920b, {40,24}*1920c, {40,24}*1920d, {80,12}*1920a, {20,48}*1920a, {80,12}*1920b, {20,48}*1920b, {40,12}*1920b, {20,24}*1920b, {20,12}*1920a, {10,96}*1920, {160,6}*1920, {40,6}*1920a, {40,12}*1920e, {40,12}*1920f, {40,6}*1920b, {20,6}*1920a, {40,6}*1920c, {20,24}*1920c, {20,24}*1920d, {40,6}*1920d, {20,6}*1920b, {20,12}*1920b, {20,12}*1920c, {40,12}*1920g, {40,12}*1920h, {20,24}*1920e, {20,24}*1920f, {10,12}*1920a
Permutation Representation (GAP) :

s0 := ( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)
(27,28)(29,30);;
s1 := ( 1, 5)( 2, 9)( 3,13)( 4,11)( 6,15)( 7,19)( 8,17)(10,21)(12,25)(14,23)
(18,29)(20,27)(24,26)(28,30);;
s2 := ( 1, 7)( 2, 3)( 4, 8)( 5,17)( 6,18)( 9,11)(10,12)(13,19)(14,20)(15,27)
(16,28)(21,23)(22,24)(25,29)(26,30);;
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*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(30)!( 5, 6)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)
(25,26)(27,28)(29,30);
s1 := Sym(30)!( 1, 5)( 2, 9)( 3,13)( 4,11)( 6,15)( 7,19)( 8,17)(10,21)(12,25)
(14,23)(18,29)(20,27)(24,26)(28,30);
s2 := Sym(30)!( 1, 7)( 2, 3)( 4, 8)( 5,17)( 6,18)( 9,11)(10,12)(13,19)(14,20)
(15,27)(16,28)(21,23)(22,24)(25,29)(26,30);
poly := sub<Sym(30)|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*s0*s1*s0*s1 >;

References : None.
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