Polytope of Type {6,10}

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

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

Permutation Representation (Magma) :

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

References : None.
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