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# Polytope of Type {12,24}

Atlas Canonical Name : {12,24}*576a
if this polytope has a name.
Group : SmallGroup(576,2279)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 12, 144, 24
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{12,24,2} of size 1152
Vertex Figure Of :
{2,12,24} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {12,12}*288c
3-fold quotients : {12,8}*192b
4-fold quotients : {12,6}*144b, {6,12}*144c
6-fold quotients : {12,4}*96a
8-fold quotients : {6,6}*72c
9-fold quotients : {4,8}*64b
12-fold quotients : {12,2}*48, {6,4}*48a
16-fold quotients : {3,6}*36
18-fold quotients : {4,4}*32
24-fold quotients : {6,2}*24
36-fold quotients : {2,4}*16, {4,2}*16
48-fold quotients : {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,24}*1152c, {24,24}*1152c, {24,24}*1152l
3-fold covers : {36,24}*1728a, {12,24}*1728a, {12,24}*1728p
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 19, 28)( 20, 30)( 21, 29)( 22, 34)( 23, 36)( 24, 35)( 25, 31)( 26, 33)
( 27, 32)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)
( 68, 72)( 69, 71)( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)
( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)( 86,144)
( 87,143)( 88,139)( 89,141)( 90,140)( 91,109)( 92,111)( 93,110)( 94,115)
( 95,117)( 96,116)( 97,112)( 98,114)( 99,113)(100,118)(101,120)(102,119)
(103,124)(104,126)(105,125)(106,121)(107,123)(108,122);;
s1 := (  1, 77)(  2, 76)(  3, 78)(  4, 74)(  5, 73)(  6, 75)(  7, 80)(  8, 79)
(  9, 81)( 10, 86)( 11, 85)( 12, 87)( 13, 83)( 14, 82)( 15, 84)( 16, 89)
( 17, 88)( 18, 90)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)( 24,102)
( 25,107)( 26,106)( 27,108)( 28, 95)( 29, 94)( 30, 96)( 31, 92)( 32, 91)
( 33, 93)( 34, 98)( 35, 97)( 36, 99)( 37,122)( 38,121)( 39,123)( 40,119)
( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,113)( 47,112)( 48,114)
( 49,110)( 50,109)( 51,111)( 52,116)( 53,115)( 54,117)( 55,131)( 56,130)
( 57,132)( 58,128)( 59,127)( 60,129)( 61,134)( 62,133)( 63,135)( 64,140)
( 65,139)( 66,141)( 67,137)( 68,136)( 69,138)( 70,143)( 71,142)( 72,144);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 19, 28)( 20, 30)
( 21, 29)( 22, 31)( 23, 33)( 24, 32)( 25, 34)( 26, 36)( 27, 35)( 38, 39)
( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 55, 64)( 56, 66)( 57, 65)
( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)( 63, 71)( 73, 91)( 74, 93)
( 75, 92)( 76, 94)( 77, 96)( 78, 95)( 79, 97)( 80, 99)( 81, 98)( 82,100)
( 83,102)( 84,101)( 85,103)( 86,105)( 87,104)( 88,106)( 89,108)( 90,107)
(109,127)(110,129)(111,128)(112,130)(113,132)(114,131)(115,133)(116,135)
(117,134)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)(124,142)
(125,144)(126,143);;
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, s2*s0*s1*s2*s1*s2*s0*s1*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(144)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 19, 28)( 20, 30)( 21, 29)( 22, 34)( 23, 36)( 24, 35)( 25, 31)
( 26, 33)( 27, 32)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)
( 67, 70)( 68, 72)( 69, 71)( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)
( 78,134)( 79,130)( 80,132)( 81,131)( 82,136)( 83,138)( 84,137)( 85,142)
( 86,144)( 87,143)( 88,139)( 89,141)( 90,140)( 91,109)( 92,111)( 93,110)
( 94,115)( 95,117)( 96,116)( 97,112)( 98,114)( 99,113)(100,118)(101,120)
(102,119)(103,124)(104,126)(105,125)(106,121)(107,123)(108,122);
s1 := Sym(144)!(  1, 77)(  2, 76)(  3, 78)(  4, 74)(  5, 73)(  6, 75)(  7, 80)
(  8, 79)(  9, 81)( 10, 86)( 11, 85)( 12, 87)( 13, 83)( 14, 82)( 15, 84)
( 16, 89)( 17, 88)( 18, 90)( 19,104)( 20,103)( 21,105)( 22,101)( 23,100)
( 24,102)( 25,107)( 26,106)( 27,108)( 28, 95)( 29, 94)( 30, 96)( 31, 92)
( 32, 91)( 33, 93)( 34, 98)( 35, 97)( 36, 99)( 37,122)( 38,121)( 39,123)
( 40,119)( 41,118)( 42,120)( 43,125)( 44,124)( 45,126)( 46,113)( 47,112)
( 48,114)( 49,110)( 50,109)( 51,111)( 52,116)( 53,115)( 54,117)( 55,131)
( 56,130)( 57,132)( 58,128)( 59,127)( 60,129)( 61,134)( 62,133)( 63,135)
( 64,140)( 65,139)( 66,141)( 67,137)( 68,136)( 69,138)( 70,143)( 71,142)
( 72,144);
s2 := Sym(144)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 19, 28)
( 20, 30)( 21, 29)( 22, 31)( 23, 33)( 24, 32)( 25, 34)( 26, 36)( 27, 35)
( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 55, 64)( 56, 66)
( 57, 65)( 58, 67)( 59, 69)( 60, 68)( 61, 70)( 62, 72)( 63, 71)( 73, 91)
( 74, 93)( 75, 92)( 76, 94)( 77, 96)( 78, 95)( 79, 97)( 80, 99)( 81, 98)
( 82,100)( 83,102)( 84,101)( 85,103)( 86,105)( 87,104)( 88,106)( 89,108)
( 90,107)(109,127)(110,129)(111,128)(112,130)(113,132)(114,131)(115,133)
(116,135)(117,134)(118,136)(119,138)(120,137)(121,139)(122,141)(123,140)
(124,142)(125,144)(126,143);
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, s2*s0*s1*s2*s1*s2*s0*s1*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