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# Polytope of Type {24,12}

Atlas Canonical Name : {24,12}*576e
if this polytope has a name.
Group : SmallGroup(576,2897)
Rank : 3
Schlafli Type : {24,12}
Number of vertices, edges, etc : 24, 144, 12
Order of s0s1s2 : 24
Order of s0s1s2s1 : 4
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 : {12,12}*288a
3-fold quotients : {24,4}*192b, {8,12}*192b
4-fold quotients : {6,12}*144a, {12,6}*144a
6-fold quotients : {4,12}*96a, {12,4}*96a
8-fold quotients : {6,6}*72a
9-fold quotients : {8,4}*64b
12-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
18-fold quotients : {4,4}*32
24-fold quotients : {2,6}*24, {6,2}*24
36-fold quotients : {2,4}*16, {4,2}*16
48-fold quotients : {2,3}*12, {3,2}*12
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {24,12}*1152b, {24,24}*1152i, {24,24}*1152k
3-fold covers : {72,12}*1728c, {24,36}*1728d, {24,12}*1728f, {24,12}*1728p
Permutation Representation (GAP) :
```s0 := (  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)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 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)
(109,136)(110,138)(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)
(117,143)(118,127)(119,129)(120,128)(121,130)(122,132)(123,131)(124,133)
(125,135)(126,134);;
s1 := (  1, 74)(  2, 73)(  3, 75)(  4, 80)(  5, 79)(  6, 81)(  7, 77)(  8, 76)
(  9, 78)( 10, 83)( 11, 82)( 12, 84)( 13, 89)( 14, 88)( 15, 90)( 16, 86)
( 17, 85)( 18, 87)( 19,101)( 20,100)( 21,102)( 22,107)( 23,106)( 24,108)
( 25,104)( 26,103)( 27,105)( 28, 92)( 29, 91)( 30, 93)( 31, 98)( 32, 97)
( 33, 99)( 34, 95)( 35, 94)( 36, 96)( 37,110)( 38,109)( 39,111)( 40,116)
( 41,115)( 42,117)( 43,113)( 44,112)( 45,114)( 46,119)( 47,118)( 48,120)
( 49,125)( 50,124)( 51,126)( 52,122)( 53,121)( 54,123)( 55,137)( 56,136)
( 57,138)( 58,143)( 59,142)( 60,144)( 61,140)( 62,139)( 63,141)( 64,128)
( 65,127)( 66,129)( 67,134)( 68,133)( 69,135)( 70,131)( 71,130)( 72,132);;
s2 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 31)( 20, 32)
( 21, 33)( 22, 28)( 23, 29)( 24, 30)( 25, 34)( 26, 35)( 27, 36)( 37, 40)
( 38, 41)( 39, 42)( 46, 49)( 47, 50)( 48, 51)( 55, 67)( 56, 68)( 57, 69)
( 58, 64)( 59, 65)( 60, 66)( 61, 70)( 62, 71)( 63, 72)( 73,112)( 74,113)
( 75,114)( 76,109)( 77,110)( 78,111)( 79,115)( 80,116)( 81,117)( 82,121)
( 83,122)( 84,123)( 85,118)( 86,119)( 87,120)( 88,124)( 89,125)( 90,126)
( 91,139)( 92,140)( 93,141)( 94,136)( 95,137)( 96,138)( 97,142)( 98,143)
( 99,144)(100,130)(101,131)(102,132)(103,127)(104,128)(105,129)(106,133)
(107,134)(108,135);;
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*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*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,
s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0 ];;
poly := F / rels;;

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

```
References : None.
to this polytope