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# Polytope of Type {3,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*432a
if this polytope has a name.
Group : SmallGroup(432,301)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 3, 9, 36, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,12,2} of size 864
{3,6,12,4} of size 1728
{3,6,12,4} of size 1728
{3,6,12,4} of size 1728
Vertex Figure Of :
{2,3,6,12} of size 864
{4,3,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,6}*216a
3-fold quotients : {3,2,12}*144
4-fold quotients : {3,6,3}*108
6-fold quotients : {3,2,6}*72
9-fold quotients : {3,2,4}*48
12-fold quotients : {3,2,3}*36
18-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,24}*864a, {6,6,12}*864a
3-fold covers : {9,6,12}*1296a, {3,6,36}*1296a, {3,6,12}*1296a, {3,6,12}*1296b, {3,6,12}*1296c
4-fold covers : {3,6,48}*1728a, {12,6,12}*1728a, {6,12,12}*1728a, {6,6,24}*1728a, {3,12,12}*1728a
Permutation Representation (GAP) :
```s0 := (  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);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)( 71, 72)
( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)( 89, 90)
( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)(107,108);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)(  8, 17)
(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)( 31, 41)
( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)( 52, 54)
( 55, 91)( 56, 93)( 57, 92)( 58, 95)( 59, 94)( 60, 96)( 61, 99)( 62, 98)
( 63, 97)( 64, 82)( 65, 84)( 66, 83)( 67, 86)( 68, 85)( 69, 87)( 70, 90)
( 71, 89)( 72, 88)( 73,100)( 74,102)( 75,101)( 76,104)( 77,103)( 78,105)
( 79,108)( 80,107)( 81,106);;
s3 := (  1, 55)(  2, 57)(  3, 56)(  4, 58)(  5, 60)(  6, 59)(  7, 61)(  8, 63)
(  9, 62)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)( 16, 79)
( 17, 81)( 18, 80)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)( 24, 68)
( 25, 70)( 26, 72)( 27, 71)( 28, 82)( 29, 84)( 30, 83)( 31, 85)( 32, 87)
( 33, 86)( 34, 88)( 35, 90)( 36, 89)( 37,100)( 38,102)( 39,101)( 40,103)
( 41,105)( 42,104)( 43,106)( 44,108)( 45,107)( 46, 91)( 47, 93)( 48, 92)
( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98);;
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*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := 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);
s1 := Sym(108)!(  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)
( 17, 18)( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)
( 53, 54)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 67)( 65, 69)( 66, 68)
( 71, 72)( 73, 76)( 74, 78)( 75, 77)( 80, 81)( 82, 85)( 83, 87)( 84, 86)
( 89, 90)( 91, 94)( 92, 96)( 93, 95)( 98, 99)(100,103)(101,105)(102,104)
(107,108);
s2 := Sym(108)!(  1, 10)(  2, 12)(  3, 11)(  4, 14)(  5, 13)(  6, 15)(  7, 18)
(  8, 17)(  9, 16)( 20, 21)( 22, 23)( 25, 27)( 28, 37)( 29, 39)( 30, 38)
( 31, 41)( 32, 40)( 33, 42)( 34, 45)( 35, 44)( 36, 43)( 47, 48)( 49, 50)
( 52, 54)( 55, 91)( 56, 93)( 57, 92)( 58, 95)( 59, 94)( 60, 96)( 61, 99)
( 62, 98)( 63, 97)( 64, 82)( 65, 84)( 66, 83)( 67, 86)( 68, 85)( 69, 87)
( 70, 90)( 71, 89)( 72, 88)( 73,100)( 74,102)( 75,101)( 76,104)( 77,103)
( 78,105)( 79,108)( 80,107)( 81,106);
s3 := Sym(108)!(  1, 55)(  2, 57)(  3, 56)(  4, 58)(  5, 60)(  6, 59)(  7, 61)
(  8, 63)(  9, 62)( 10, 73)( 11, 75)( 12, 74)( 13, 76)( 14, 78)( 15, 77)
( 16, 79)( 17, 81)( 18, 80)( 19, 64)( 20, 66)( 21, 65)( 22, 67)( 23, 69)
( 24, 68)( 25, 70)( 26, 72)( 27, 71)( 28, 82)( 29, 84)( 30, 83)( 31, 85)
( 32, 87)( 33, 86)( 34, 88)( 35, 90)( 36, 89)( 37,100)( 38,102)( 39,101)
( 40,103)( 41,105)( 42,104)( 43,106)( 44,108)( 45,107)( 46, 91)( 47, 93)
( 48, 92)( 49, 94)( 50, 96)( 51, 95)( 52, 97)( 53, 99)( 54, 98);
poly := sub<Sym(108)|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*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope