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# Polytope of Type {24,12}

Atlas Canonical Name : {24,12}*576d
if this polytope has a name.
Group : SmallGroup(576,2829)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 24, 144, 12
Order of s0s1s2 : 24
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{24,12,2} of size 1152
Vertex Figure Of :
{2,24,12} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {24,6}*288b, {12,12}*288c
3-fold quotients : {24,4}*192a
4-fold quotients : {12,6}*144b, {6,12}*144c
6-fold quotients : {12,4}*96a, {24,2}*96
8-fold quotients : {6,6}*72c
9-fold quotients : {8,4}*64a
12-fold quotients : {12,2}*48, {6,4}*48a
16-fold quotients : {3,6}*36
18-fold quotients : {4,4}*32, {8,2}*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 : {24,12}*1152a, {24,24}*1152c, {24,24}*1152e, {48,12}*1152a, {48,12}*1152d
3-fold covers : {72,12}*1728b, {24,12}*1728c, {24,12}*1728o
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)
( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)( 44, 51)
( 45, 50)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)( 61, 67)
( 62, 69)( 63, 68)( 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, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)( 24, 93)
( 25, 98)( 26, 97)( 27, 99)( 28,104)( 29,103)( 30,105)( 31,101)( 32,100)
( 33,102)( 34,107)( 35,106)( 36,108)( 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)( 55,140)( 56,139)
( 57,141)( 58,137)( 59,136)( 60,138)( 61,143)( 62,142)( 63,144)( 64,131)
( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)( 72,135);;
s2 := (  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)
( 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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*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*s0*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*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)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 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, 95)( 20, 94)( 21, 96)( 22, 92)( 23, 91)
( 24, 93)( 25, 98)( 26, 97)( 27, 99)( 28,104)( 29,103)( 30,105)( 31,101)
( 32,100)( 33,102)( 34,107)( 35,106)( 36,108)( 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)( 55,140)
( 56,139)( 57,141)( 58,137)( 59,136)( 60,138)( 61,143)( 62,142)( 63,144)
( 64,131)( 65,130)( 66,132)( 67,128)( 68,127)( 69,129)( 70,134)( 71,133)
( 72,135);
s2 := Sym(144)!(  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)( 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*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*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*s0*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*s0*s1*s0*s1*s0*s1 >;

```
References : None.
to this polytope