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# Polytope of Type {18,8,2}

Atlas Canonical Name : {18,8,2}*576
if this polytope has a name.
Group : SmallGroup(576,1738)
Rank : 4
Schlafli Type : {18,8,2}
Number of vertices, edges, etc : 18, 72, 8, 2
Order of s0s1s2s3 : 72
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{18,8,2,2} of size 1152
{18,8,2,3} of size 1728
Vertex Figure Of :
{2,18,8,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,4,2}*288a
3-fold quotients : {6,8,2}*192
4-fold quotients : {18,2,2}*144
6-fold quotients : {6,4,2}*96a
8-fold quotients : {9,2,2}*72
9-fold quotients : {2,8,2}*64
12-fold quotients : {6,2,2}*48
18-fold quotients : {2,4,2}*32
24-fold quotients : {3,2,2}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {18,8,4}*1152a, {36,8,2}*1152a, {18,16,2}*1152
3-fold covers : {54,8,2}*1728, {18,24,2}*1728a, {18,8,6}*1728, {18,24,2}*1728b
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)
(23,25)(24,27)(29,30)(31,35)(32,34)(33,36)(38,39)(40,44)(41,43)(42,45)(47,48)
(49,53)(50,52)(51,54)(56,57)(58,62)(59,61)(60,63)(65,66)(67,71)(68,70)
(69,72);;
s1 := ( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)(20,33)
(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36)(37,58)(38,60)(39,59)(40,55)
(41,57)(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)(51,65)
(52,71)(53,70)(54,72);;
s2 := ( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)(10,46)
(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)(21,66)
(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)(32,59)
(33,60)(34,61)(35,62)(36,63);;
s3 := (73,74);;
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(74)!( 2, 3)( 4, 8)( 5, 7)( 6, 9)(11,12)(13,17)(14,16)(15,18)(20,21)
(22,26)(23,25)(24,27)(29,30)(31,35)(32,34)(33,36)(38,39)(40,44)(41,43)(42,45)
(47,48)(49,53)(50,52)(51,54)(56,57)(58,62)(59,61)(60,63)(65,66)(67,71)(68,70)
(69,72);
s1 := Sym(74)!( 1, 4)( 2, 6)( 3, 5)( 7, 8)(10,13)(11,15)(12,14)(16,17)(19,31)
(20,33)(21,32)(22,28)(23,30)(24,29)(25,35)(26,34)(27,36)(37,58)(38,60)(39,59)
(40,55)(41,57)(42,56)(43,62)(44,61)(45,63)(46,67)(47,69)(48,68)(49,64)(50,66)
(51,65)(52,71)(53,70)(54,72);
s2 := Sym(74)!( 1,37)( 2,38)( 3,39)( 4,40)( 5,41)( 6,42)( 7,43)( 8,44)( 9,45)
(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,64)(20,65)
(21,66)(22,67)(23,68)(24,69)(25,70)(26,71)(27,72)(28,55)(29,56)(30,57)(31,58)
(32,59)(33,60)(34,61)(35,62)(36,63);
s3 := Sym(74)!(73,74);
poly := sub<Sym(74)|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*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

```

to this polytope