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# Polytope of Type {12,6}

Atlas Canonical Name : {12,6}*432a
if this polytope has a name.
Group : SmallGroup(432,301)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 36, 108, 18
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{12,6,2} of size 864
{12,6,3} of size 1296
{12,6,4} of size 1728
Vertex Figure Of :
{2,12,6} of size 864
{4,12,6} of size 1728
{4,12,6} of size 1728
{4,12,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*216c
3-fold quotients : {12,6}*144b
4-fold quotients : {3,6}*108
6-fold quotients : {6,6}*72c
9-fold quotients : {12,2}*48
12-fold quotients : {3,6}*36
18-fold quotients : {6,2}*24
27-fold quotients : {4,2}*16
36-fold quotients : {3,2}*12
54-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6}*864a, {12,12}*864a
3-fold covers : {36,6}*1296a, {36,6}*1296c, {36,6}*1296d, {36,6}*1296e, {12,18}*1296d, {12,6}*1296c, {12,6}*1296h
4-fold covers : {48,6}*1728a, {12,24}*1728a, {12,12}*1728a, {12,24}*1728b, {24,12}*1728c, {24,12}*1728e, {12,12}*1728i, {12,6}*1728a
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)( 62, 86)
( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)( 70,103)
( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)
( 79, 94)( 80, 95)( 81, 96);;
s1 := (  1, 77)(  2, 78)(  3, 76)(  4, 75)(  5, 73)(  6, 74)(  7, 79)(  8, 80)
(  9, 81)( 10, 68)( 11, 69)( 12, 67)( 13, 66)( 14, 64)( 15, 65)( 16, 70)
( 17, 71)( 18, 72)( 19, 59)( 20, 60)( 21, 58)( 22, 57)( 23, 55)( 24, 56)
( 25, 61)( 26, 62)( 27, 63)( 28,104)( 29,105)( 30,103)( 31,102)( 32,100)
( 33,101)( 34,106)( 35,107)( 36,108)( 37, 95)( 38, 96)( 39, 94)( 40, 93)
( 41, 91)( 42, 92)( 43, 97)( 44, 98)( 45, 99)( 46, 86)( 47, 87)( 48, 85)
( 49, 84)( 50, 82)( 51, 83)( 52, 88)( 53, 89)( 54, 90);;
s2 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107);;
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*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(108)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 31, 34)( 32, 35)( 33, 36)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)( 61, 85)
( 62, 86)( 63, 87)( 64,100)( 65,101)( 66,102)( 67,106)( 68,107)( 69,108)
( 70,103)( 71,104)( 72,105)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)
( 78, 99)( 79, 94)( 80, 95)( 81, 96);
s1 := Sym(108)!(  1, 77)(  2, 78)(  3, 76)(  4, 75)(  5, 73)(  6, 74)(  7, 79)
(  8, 80)(  9, 81)( 10, 68)( 11, 69)( 12, 67)( 13, 66)( 14, 64)( 15, 65)
( 16, 70)( 17, 71)( 18, 72)( 19, 59)( 20, 60)( 21, 58)( 22, 57)( 23, 55)
( 24, 56)( 25, 61)( 26, 62)( 27, 63)( 28,104)( 29,105)( 30,103)( 31,102)
( 32,100)( 33,101)( 34,106)( 35,107)( 36,108)( 37, 95)( 38, 96)( 39, 94)
( 40, 93)( 41, 91)( 42, 92)( 43, 97)( 44, 98)( 45, 99)( 46, 86)( 47, 87)
( 48, 85)( 49, 84)( 50, 82)( 51, 83)( 52, 88)( 53, 89)( 54, 90);
s2 := Sym(108)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107);
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, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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 >;

```
References : None.
to this polytope