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# Polytope of Type {12,24}

Atlas Canonical Name : {12,24}*576c
Also Known As : {12,24|2}. if this polytope has another name.
Group : SmallGroup(576,2829)
Rank : 3
Schlafli Type : {12,24}
Number of vertices, edges, etc : 12, 144, 24
Order of s0s1s2 : 24
Order of s0s1s2s1 : 2
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 : {6,24}*288a, {12,12}*288a
3-fold quotients : {4,24}*192a, {12,8}*192a
4-fold quotients : {6,12}*144a, {12,6}*144a
6-fold quotients : {4,12}*96a, {12,4}*96a, {2,24}*96, {6,8}*96
8-fold quotients : {6,6}*72a
9-fold quotients : {4,8}*64a
12-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
18-fold quotients : {4,4}*32, {2,8}*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 : {12,24}*1152b, {24,24}*1152b, {24,24}*1152i, {12,48}*1152b, {12,48}*1152e
3-fold covers : {12,72}*1728a, {36,24}*1728c, {12,24}*1728d, {12,24}*1728o
Permutation Representation (GAP) :
```s0 := (  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);;
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, 92)( 20, 91)( 21, 93)( 22, 98)( 23, 97)( 24, 99)
( 25, 95)( 26, 94)( 27, 96)( 28,101)( 29,100)( 30,102)( 31,107)( 32,106)
( 33,108)( 34,104)( 35,103)( 36,105)( 37,119)( 38,118)( 39,120)( 40,125)
( 41,124)( 42,126)( 43,122)( 44,121)( 45,123)( 46,110)( 47,109)( 48,111)
( 49,116)( 50,115)( 51,117)( 52,113)( 53,112)( 54,114)( 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, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 49)( 38, 50)( 39, 51)( 40, 46)
( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)( 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)
( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)( 97,133)( 98,134)
( 99,135)(100,139)(101,140)(102,141)(103,136)(104,137)(105,138)(106,142)
(107,143)(108,144);;
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*s2*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,
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*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)(  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);
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, 92)( 20, 91)( 21, 93)( 22, 98)( 23, 97)
( 24, 99)( 25, 95)( 26, 94)( 27, 96)( 28,101)( 29,100)( 30,102)( 31,107)
( 32,106)( 33,108)( 34,104)( 35,103)( 36,105)( 37,119)( 38,118)( 39,120)
( 40,125)( 41,124)( 42,126)( 43,122)( 44,121)( 45,123)( 46,110)( 47,109)
( 48,111)( 49,116)( 50,115)( 51,117)( 52,113)( 53,112)( 54,114)( 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, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 49)( 38, 50)( 39, 51)
( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)( 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)( 91,130)( 92,131)( 93,132)( 94,127)( 95,128)( 96,129)( 97,133)
( 98,134)( 99,135)(100,139)(101,140)(102,141)(103,136)(104,137)(105,138)
(106,142)(107,143)(108,144);
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*s2*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,
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*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