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# Polytope of Type {18,4}

Atlas Canonical Name : {18,4}*144b
if this polytope has a name.
Group : SmallGroup(144,109)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 18, 36, 4
Order of s0s1s2 : 18
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{18,4,2} of size 288
{18,4,4} of size 1152
Vertex Figure Of :
{2,18,4} of size 288
{4,18,4} of size 576
{4,18,4} of size 576
{4,18,4} of size 576
{6,18,4} of size 864
{6,18,4} of size 864
{8,18,4} of size 1152
{4,18,4} of size 1152
{6,18,4} of size 1296
{6,18,4} of size 1296
{6,18,4} of size 1296
{10,18,4} of size 1440
{12,18,4} of size 1728
{12,18,4} of size 1728
{12,18,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,4}*72
3-fold quotients : {6,4}*48c
6-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,4}*288b, {36,4}*288c, {18,4}*288
3-fold covers : {54,4}*432b
4-fold covers : {18,4}*576a, {72,4}*576c, {72,4}*576d, {36,4}*576b, {18,4}*576b, {36,4}*576c, {18,8}*576b, {18,8}*576c
5-fold covers : {90,4}*720b
6-fold covers : {108,4}*864b, {108,4}*864c, {54,4}*864, {18,12}*864a, {18,12}*864b
7-fold covers : {126,4}*1008b
8-fold covers : {36,4}*1152b, {36,4}*1152c, {18,8}*1152b, {18,8}*1152c, {144,4}*1152c, {144,4}*1152d, {36,4}*1152d, {36,8}*1152e, {36,8}*1152f, {18,4}*1152a, {18,8}*1152d, {18,8}*1152e, {18,8}*1152f, {36,8}*1152g, {36,8}*1152h, {72,4}*1152c, {72,4}*1152d, {18,8}*1152g, {36,4}*1152e, {72,4}*1152e, {18,4}*1152b, {72,4}*1152f
9-fold covers : {162,4}*1296b
10-fold covers : {180,4}*1440b, {180,4}*1440c, {18,20}*1440, {90,4}*1440
11-fold covers : {198,4}*1584b
12-fold covers : {54,4}*1728a, {216,4}*1728c, {216,4}*1728d, {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c, {36,12}*1728e, {36,12}*1728f, {18,12}*1728c, {36,12}*1728g, {18,24}*1728b, {18,24}*1728c, {18,24}*1728d, {18,24}*1728e, {18,12}*1728d, {36,12}*1728h
13-fold covers : {234,4}*1872b
Permutation Representation (GAP) :
```s0 := ( 2, 3)( 5, 9)( 6,11)( 7,10)( 8,12)(13,29)(14,31)(15,30)(16,32)(17,25)
(18,27)(19,26)(20,28)(21,33)(22,35)(23,34)(24,36)(38,39)(41,45)(42,47)(43,46)
(44,48)(49,65)(50,67)(51,66)(52,68)(53,61)(54,63)(55,62)(56,64)(57,69)(58,71)
(59,70)(60,72);;
s1 := ( 1,49)( 2,50)( 3,52)( 4,51)( 5,57)( 6,58)( 7,60)( 8,59)( 9,53)(10,54)
(11,56)(12,55)(13,37)(14,38)(15,40)(16,39)(17,45)(18,46)(19,48)(20,47)(21,41)
(22,42)(23,44)(24,43)(25,65)(26,66)(27,68)(28,67)(29,61)(30,62)(31,64)(32,63)
(33,69)(34,70)(35,72)(36,71);;
s2 := ( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)
(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)
(42,43)(45,48)(46,47)(49,52)(50,51)(53,56)(54,55)(57,60)(58,59)(61,64)(62,63)
(65,68)(66,67)(69,72)(70,71);;
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*s1*s0*s2*s1*s2*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*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope