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# Polytope of Type {12,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,6}*864c
if this polytope has a name.
Group : SmallGroup(864,4368)
Rank : 4
Schlafli Type : {12,6,6}
Number of vertices, edges, etc : 12, 36, 18, 6
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,6,6,2} of size 1728
Vertex Figure Of :
{2,12,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6,6}*432e
3-fold quotients : {12,6,2}*288a, {4,6,6}*288b
6-fold quotients : {2,6,6}*144c, {6,6,2}*144a
9-fold quotients : {12,2,2}*96, {4,6,2}*96a
12-fold quotients : {2,3,6}*72
18-fold quotients : {2,6,2}*48, {6,2,2}*48
27-fold quotients : {4,2,2}*32
36-fold quotients : {2,3,2}*24, {3,2,2}*24
54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,6,6}*1728c, {12,6,12}*1728d, {12,12,6}*1728d
Permutation Representation (GAP) :
```s0 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)
( 61, 88)( 62, 89)( 63, 90)( 64,100)( 65,101)( 66,102)( 67,103)( 68,104)
( 69,105)( 70,106)( 71,107)( 72,108)( 73, 91)( 74, 92)( 75, 93)( 76, 94)
( 77, 95)( 78, 96)( 79, 97)( 80, 98)( 81, 99);;
s1 := (  1, 64)(  2, 66)(  3, 65)(  4, 70)(  5, 72)(  6, 71)(  7, 67)(  8, 69)
(  9, 68)( 10, 55)( 11, 57)( 12, 56)( 13, 61)( 14, 63)( 15, 62)( 16, 58)
( 17, 60)( 18, 59)( 19, 73)( 20, 75)( 21, 74)( 22, 79)( 23, 81)( 24, 80)
( 25, 76)( 26, 78)( 27, 77)( 28, 91)( 29, 93)( 30, 92)( 31, 97)( 32, 99)
( 33, 98)( 34, 94)( 35, 96)( 36, 95)( 37, 82)( 38, 84)( 39, 83)( 40, 88)
( 41, 90)( 42, 89)( 43, 85)( 44, 87)( 45, 86)( 46,100)( 47,102)( 48,101)
( 49,106)( 50,108)( 51,107)( 52,103)( 53,105)( 54,104);;
s2 := (  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 14)( 11, 13)( 12, 15)( 16, 17)
( 19, 23)( 20, 22)( 21, 24)( 25, 26)( 28, 32)( 29, 31)( 30, 33)( 34, 35)
( 37, 41)( 38, 40)( 39, 42)( 43, 44)( 46, 50)( 47, 49)( 48, 51)( 52, 53)
( 55, 59)( 56, 58)( 57, 60)( 61, 62)( 64, 68)( 65, 67)( 66, 69)( 70, 71)
( 73, 77)( 74, 76)( 75, 78)( 79, 80)( 82, 86)( 83, 85)( 84, 87)( 88, 89)
( 91, 95)( 92, 94)( 93, 96)( 97, 98)(100,104)(101,103)(102,105)(106,107);;
s3 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108);;
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*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s3*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope