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# Polytope of Type {6,12}

Atlas Canonical Name : {6,12}*576e
if this polytope has a name.
Group : SmallGroup(576,8355)
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}*288b
3-fold quotients : {6,12}*192a
4-fold quotients : {6,12}*144c, {3,6}*144
6-fold quotients : {6,6}*96
8-fold quotients : {6,6}*72c
12-fold quotients : {6,4}*48a, {3,6}*48, {6,3}*48
16-fold quotients : {3,6}*36
24-fold quotients : {3,3}*24, {6,2}*24
36-fold quotients : {2,4}*16
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,12}*1152l, {6,24}*1152j, {6,12}*1152e, {12,12}*1152q, {6,24}*1152m
3-fold covers : {18,12}*1728b, {6,12}*1728e, {6,12}*1728h
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 25)( 14, 27)( 15, 26)
( 16, 28)( 17, 33)( 18, 35)( 19, 34)( 20, 36)( 21, 29)( 22, 31)( 23, 30)
( 24, 32)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 61)( 50, 63)
( 51, 62)( 52, 64)( 53, 69)( 54, 71)( 55, 70)( 56, 72)( 57, 65)( 58, 67)
( 59, 66)( 60, 68)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85, 97)
( 86, 99)( 87, 98)( 88,100)( 89,105)( 90,107)( 91,106)( 92,108)( 93,101)
( 94,103)( 95,102)( 96,104)(110,111)(113,117)(114,119)(115,118)(116,120)
(121,133)(122,135)(123,134)(124,136)(125,141)(126,143)(127,142)(128,144)
(129,137)(130,139)(131,138)(132,140);;
s1 := (  1, 89)(  2, 90)(  3, 92)(  4, 91)(  5, 85)(  6, 86)(  7, 88)(  8, 87)
(  9, 93)( 10, 94)( 11, 96)( 12, 95)( 13, 77)( 14, 78)( 15, 80)( 16, 79)
( 17, 73)( 18, 74)( 19, 76)( 20, 75)( 21, 81)( 22, 82)( 23, 84)( 24, 83)
( 25,101)( 26,102)( 27,104)( 28,103)( 29, 97)( 30, 98)( 31,100)( 32, 99)
( 33,105)( 34,106)( 35,108)( 36,107)( 37,125)( 38,126)( 39,128)( 40,127)
( 41,121)( 42,122)( 43,124)( 44,123)( 45,129)( 46,130)( 47,132)( 48,131)
( 49,113)( 50,114)( 51,116)( 52,115)( 53,109)( 54,110)( 55,112)( 56,111)
( 57,117)( 58,118)( 59,120)( 60,119)( 61,137)( 62,138)( 63,140)( 64,139)
( 65,133)( 66,134)( 67,136)( 68,135)( 69,141)( 70,142)( 71,144)( 72,143);;
s2 := (  1,  4)(  5,  8)(  9, 12)( 13, 28)( 14, 26)( 15, 27)( 16, 25)( 17, 32)
( 18, 30)( 19, 31)( 20, 29)( 21, 36)( 22, 34)( 23, 35)( 24, 33)( 37, 40)
( 41, 44)( 45, 48)( 49, 64)( 50, 62)( 51, 63)( 52, 61)( 53, 68)( 54, 66)
( 55, 67)( 56, 65)( 57, 72)( 58, 70)( 59, 71)( 60, 69)( 73,112)( 74,110)
( 75,111)( 76,109)( 77,116)( 78,114)( 79,115)( 80,113)( 81,120)( 82,118)
( 83,119)( 84,117)( 85,136)( 86,134)( 87,135)( 88,133)( 89,140)( 90,138)
( 91,139)( 92,137)( 93,144)( 94,142)( 95,143)( 96,141)( 97,124)( 98,122)
( 99,123)(100,121)(101,128)(102,126)(103,127)(104,125)(105,132)(106,130)
(107,131)(108,129);;
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,
s2*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope