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# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*576a
if this polytope has a name.
Group : SmallGroup(576,8313)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 24, 144, 48
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 :
{6,12,2} of size 1152
Vertex Figure Of :
{2,6,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,6}*288a
3-fold quotients : {6,12}*192a
4-fold quotients : {6,12}*144b, {6,3}*144
6-fold quotients : {6,6}*96
8-fold quotients : {6,6}*72b
12-fold quotients : {2,12}*48, {3,6}*48, {6,3}*48
16-fold quotients : {6,3}*36
24-fold quotients : {3,3}*24, {2,6}*24
36-fold quotients : {2,4}*16
48-fold quotients : {2,3}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1152f, {6,12}*1152a, {6,24}*1152g, {6,24}*1152i, {12,12}*1152j
3-fold covers : {6,36}*1728a, {6,12}*1728a, {6,12}*1728h
Permutation Representation (GAP) :
```s0 := (  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)( 17, 29)
( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)( 38, 39)
( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)( 54, 67)
( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 78, 79)
( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)( 91,102)
( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)(118,119)
(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)(128,140)
(129,141)(130,143)(131,142)(132,144);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)( 31, 36)
( 32, 35)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)( 43, 60)
( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 63, 64)( 65, 69)( 66, 70)
( 67, 72)( 68, 71)( 73,121)( 74,122)( 75,124)( 76,123)( 77,129)( 78,130)
( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)( 84,127)( 85,109)( 86,110)
( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)( 92,119)( 93,113)( 94,114)
( 95,116)( 96,115)( 97,133)( 98,134)( 99,136)(100,135)(101,141)(102,142)
(103,144)(104,143)(105,137)(106,138)(107,140)(108,139);;
s2 := (  1, 80)(  2, 78)(  3, 79)(  4, 77)(  5, 76)(  6, 74)(  7, 75)(  8, 73)
(  9, 84)( 10, 82)( 11, 83)( 12, 81)( 13,104)( 14,102)( 15,103)( 16,101)
( 17,100)( 18, 98)( 19, 99)( 20, 97)( 21,108)( 22,106)( 23,107)( 24,105)
( 25, 92)( 26, 90)( 27, 91)( 28, 89)( 29, 88)( 30, 86)( 31, 87)( 32, 85)
( 33, 96)( 34, 94)( 35, 95)( 36, 93)( 37,116)( 38,114)( 39,115)( 40,113)
( 41,112)( 42,110)( 43,111)( 44,109)( 45,120)( 46,118)( 47,119)( 48,117)
( 49,140)( 50,138)( 51,139)( 52,137)( 53,136)( 54,134)( 55,135)( 56,133)
( 57,144)( 58,142)( 59,143)( 60,141)( 61,128)( 62,126)( 63,127)( 64,125)
( 65,124)( 66,122)( 67,123)( 68,121)( 69,132)( 70,130)( 71,131)( 72,129);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!(  2,  3)(  6,  7)( 10, 11)( 13, 25)( 14, 27)( 15, 26)( 16, 28)
( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 33)( 22, 35)( 23, 34)( 24, 36)
( 38, 39)( 42, 43)( 46, 47)( 49, 61)( 50, 63)( 51, 62)( 52, 64)( 53, 65)
( 54, 67)( 55, 66)( 56, 68)( 57, 69)( 58, 71)( 59, 70)( 60, 72)( 74, 75)
( 78, 79)( 82, 83)( 85, 97)( 86, 99)( 87, 98)( 88,100)( 89,101)( 90,103)
( 91,102)( 92,104)( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(114,115)
(118,119)(121,133)(122,135)(123,134)(124,136)(125,137)(126,139)(127,138)
(128,140)(129,141)(130,143)(131,142)(132,144);
s1 := Sym(144)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 27, 28)( 29, 33)( 30, 34)
( 31, 36)( 32, 35)( 37, 49)( 38, 50)( 39, 52)( 40, 51)( 41, 57)( 42, 58)
( 43, 60)( 44, 59)( 45, 53)( 46, 54)( 47, 56)( 48, 55)( 63, 64)( 65, 69)
( 66, 70)( 67, 72)( 68, 71)( 73,121)( 74,122)( 75,124)( 76,123)( 77,129)
( 78,130)( 79,132)( 80,131)( 81,125)( 82,126)( 83,128)( 84,127)( 85,109)
( 86,110)( 87,112)( 88,111)( 89,117)( 90,118)( 91,120)( 92,119)( 93,113)
( 94,114)( 95,116)( 96,115)( 97,133)( 98,134)( 99,136)(100,135)(101,141)
(102,142)(103,144)(104,143)(105,137)(106,138)(107,140)(108,139);
s2 := Sym(144)!(  1, 80)(  2, 78)(  3, 79)(  4, 77)(  5, 76)(  6, 74)(  7, 75)
(  8, 73)(  9, 84)( 10, 82)( 11, 83)( 12, 81)( 13,104)( 14,102)( 15,103)
( 16,101)( 17,100)( 18, 98)( 19, 99)( 20, 97)( 21,108)( 22,106)( 23,107)
( 24,105)( 25, 92)( 26, 90)( 27, 91)( 28, 89)( 29, 88)( 30, 86)( 31, 87)
( 32, 85)( 33, 96)( 34, 94)( 35, 95)( 36, 93)( 37,116)( 38,114)( 39,115)
( 40,113)( 41,112)( 42,110)( 43,111)( 44,109)( 45,120)( 46,118)( 47,119)
( 48,117)( 49,140)( 50,138)( 51,139)( 52,137)( 53,136)( 54,134)( 55,135)
( 56,133)( 57,144)( 58,142)( 59,143)( 60,141)( 61,128)( 62,126)( 63,127)
( 64,125)( 65,124)( 66,122)( 67,123)( 68,121)( 69,132)( 70,130)( 71,131)
( 72,129);
poly := sub<Sym(144)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*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 >;

```
References : None.
to this polytope