Questions?
See the FAQ
or other info.

# Polytope of Type {6,10,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,10,4}*480
Also Known As : {{6,10|2},{10,4|2}}. if this polytope has another name.
Group : SmallGroup(480,1097)
Rank : 4
Schlafli Type : {6,10,4}
Number of vertices, edges, etc : 6, 30, 20, 4
Order of s0s1s2s3 : 60
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,10,4,2} of size 960
{6,10,4,4} of size 1920
Vertex Figure Of :
{2,6,10,4} of size 960
{3,6,10,4} of size 1440
{4,6,10,4} of size 1920
{3,6,10,4} of size 1920
{4,6,10,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,10,2}*240
3-fold quotients : {2,10,4}*160
5-fold quotients : {6,2,4}*96
6-fold quotients : {2,10,2}*80
10-fold quotients : {3,2,4}*48, {6,2,2}*48
12-fold quotients : {2,5,2}*40
15-fold quotients : {2,2,4}*32
20-fold quotients : {3,2,2}*24
30-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,20,4}*960, {12,10,4}*960, {6,10,8}*960
3-fold covers : {18,10,4}*1440, {6,10,12}*1440, {6,30,4}*1440a, {6,30,4}*1440b
4-fold covers : {12,20,4}*1920, {6,20,8}*1920a, {6,40,4}*1920a, {6,20,8}*1920b, {6,40,4}*1920b, {6,20,4}*1920a, {12,10,8}*1920, {24,10,4}*1920, {6,10,16}*1920, {6,20,4}*1920c
Permutation Representation (GAP) :
```s0 := ( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,26)(22,27)(23,28)(24,29)(25,30)
(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60);;
s1 := ( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)(18,24)
(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)
(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59);;
s2 := ( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)(23,25)
(26,27)(28,30)(31,47)(32,46)(33,50)(34,49)(35,48)(36,52)(37,51)(38,55)(39,54)
(40,53)(41,57)(42,56)(43,60)(44,59)(45,58);;
s3 := ( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,36)( 7,37)( 8,38)( 9,39)(10,40)
(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,51)
(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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(60)!( 6,11)( 7,12)( 8,13)( 9,14)(10,15)(21,26)(22,27)(23,28)(24,29)
(25,30)(36,41)(37,42)(38,43)(39,44)(40,45)(51,56)(52,57)(53,58)(54,59)(55,60);
s1 := Sym(60)!( 1, 6)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)(16,21)(17,25)
(18,24)(19,23)(20,22)(27,30)(28,29)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)
(43,44)(46,51)(47,55)(48,54)(49,53)(50,52)(57,60)(58,59);
s2 := Sym(60)!( 1, 2)( 3, 5)( 6, 7)( 8,10)(11,12)(13,15)(16,17)(18,20)(21,22)
(23,25)(26,27)(28,30)(31,47)(32,46)(33,50)(34,49)(35,48)(36,52)(37,51)(38,55)
(39,54)(40,53)(41,57)(42,56)(43,60)(44,59)(45,58);
s3 := Sym(60)!( 1,31)( 2,32)( 3,33)( 4,34)( 5,35)( 6,36)( 7,37)( 8,38)( 9,39)
(10,40)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)
(21,51)(22,52)(23,53)(24,54)(25,55)(26,56)(27,57)(28,58)(29,59)(30,60);
poly := sub<Sym(60)|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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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.
to this polytope