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# Polytope of Type {12,18}

Atlas Canonical Name : {12,18}*432b
if this polytope has a name.
Group : SmallGroup(432,315)
Rank : 3
Schlafli Type : {12,18}
Number of vertices, edges, etc : 12, 108, 18
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,18,2} of size 864
{12,18,4} of size 1728
{12,18,4} of size 1728
Vertex Figure Of :
{2,12,18} of size 864
{4,12,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,18}*216b
3-fold quotients : {4,18}*144a, {12,6}*144c
4-fold quotients : {6,9}*108
6-fold quotients : {2,18}*72, {6,6}*72b
9-fold quotients : {4,6}*48a
12-fold quotients : {2,9}*36, {6,3}*36
18-fold quotients : {2,6}*24
27-fold quotients : {4,2}*16
36-fold quotients : {2,3}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,36}*864b, {24,18}*864b
3-fold covers : {36,18}*1296c, {12,18}*1296e, {12,54}*1296b, {12,18}*1296l
4-fold covers : {24,36}*1728a, {12,36}*1728b, {24,36}*1728b, {12,72}*1728b, {12,72}*1728d, {48,18}*1728b, {12,18}*1728b, {12,18}*1728d
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);;
s1 := (  1, 58)(  2, 60)(  3, 59)(  4, 55)(  5, 57)(  6, 56)(  7, 61)(  8, 63)
(  9, 62)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)( 16, 80)
( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)( 24, 66)
( 25, 71)( 26, 70)( 27, 72)( 28, 85)( 29, 87)( 30, 86)( 31, 82)( 32, 84)
( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,104)( 38,103)( 39,105)( 40,101)
( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)( 48, 96)
( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)( 49, 53)
( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,101)(103,107)(104,106)(105,108);;
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, s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(108)!(  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)
( 60, 90)( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)
( 68, 98)( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)
( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)( 81,105);
s1 := Sym(108)!(  1, 58)(  2, 60)(  3, 59)(  4, 55)(  5, 57)(  6, 56)(  7, 61)
(  8, 63)(  9, 62)( 10, 77)( 11, 76)( 12, 78)( 13, 74)( 14, 73)( 15, 75)
( 16, 80)( 17, 79)( 18, 81)( 19, 68)( 20, 67)( 21, 69)( 22, 65)( 23, 64)
( 24, 66)( 25, 71)( 26, 70)( 27, 72)( 28, 85)( 29, 87)( 30, 86)( 31, 82)
( 32, 84)( 33, 83)( 34, 88)( 35, 90)( 36, 89)( 37,104)( 38,103)( 39,105)
( 40,101)( 41,100)( 42,102)( 43,107)( 44,106)( 45,108)( 46, 95)( 47, 94)
( 48, 96)( 49, 92)( 50, 91)( 51, 93)( 52, 98)( 53, 97)( 54, 99);
s2 := Sym(108)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 37)( 29, 39)
( 30, 38)( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 47)
( 49, 53)( 50, 52)( 51, 54)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)
( 60, 71)( 61, 67)( 62, 69)( 63, 68)( 73, 74)( 76, 80)( 77, 79)( 78, 81)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,101)(103,107)(104,106)(105,108);
poly := sub<Sym(108)|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, s2*s0*s1*s0*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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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