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# Polytope of Type {6,24}

Atlas Canonical Name : {6,24}*576e
if this polytope has a name.
Group : SmallGroup(576,8340)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 12, 144, 48
Order of s0s1s2 : 12
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{6,24,2} of size 1152
Vertex Figure Of :
{2,6,24} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,12}*288a
3-fold quotients : {6,8}*192b
4-fold quotients : {6,12}*144d
6-fold quotients : {3,8}*96, {6,4}*96
8-fold quotients : {6,6}*72a
12-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
24-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
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}*1152k, {6,24}*1152d, {12,24}*1152v
3-fold covers : {6,72}*1728c, {18,24}*1728e, {6,24}*1728e, {6,24}*1728f
Permutation Representation (GAP) :
```s0 := (  3,  6)(  4,  5)(  7,  8)( 11, 14)( 12, 13)( 15, 16)( 19, 22)( 20, 21)
( 23, 24)( 25, 49)( 26, 50)( 27, 54)( 28, 53)( 29, 52)( 30, 51)( 31, 56)
( 32, 55)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)( 38, 59)( 39, 64)
( 40, 63)( 41, 65)( 42, 66)( 43, 70)( 44, 69)( 45, 68)( 46, 67)( 47, 72)
( 48, 71)( 73, 74)( 75, 77)( 76, 78)( 81, 82)( 83, 85)( 84, 86)( 89, 90)
( 91, 93)( 92, 94)( 97,122)( 98,121)( 99,125)(100,126)(101,123)(102,124)
(103,127)(104,128)(105,130)(106,129)(107,133)(108,134)(109,131)(110,132)
(111,135)(112,136)(113,138)(114,137)(115,141)(116,142)(117,139)(118,140)
(119,143)(120,144);;
s1 := (  1, 25)(  2, 26)(  3, 28)(  4, 27)(  5, 31)(  6, 32)(  7, 29)(  8, 30)
(  9, 41)( 10, 42)( 11, 44)( 12, 43)( 13, 47)( 14, 48)( 15, 45)( 16, 46)
( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 39)( 22, 40)( 23, 37)( 24, 38)
( 51, 52)( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)
( 62, 72)( 63, 69)( 64, 70)( 73, 98)( 74, 97)( 75, 99)( 76,100)( 77,104)
( 78,103)( 79,102)( 80,101)( 81,114)( 82,113)( 83,115)( 84,116)( 85,120)
( 86,119)( 87,118)( 88,117)( 89,106)( 90,105)( 91,107)( 92,108)( 93,112)
( 94,111)( 95,110)( 96,109)(121,122)(125,128)(126,127)(129,138)(130,137)
(131,139)(132,140)(133,144)(134,143)(135,142)(136,141);;
s2 := (  1, 87)(  2, 88)(  3, 85)(  4, 86)(  5, 84)(  6, 83)(  7, 82)(  8, 81)
(  9, 79)( 10, 80)( 11, 77)( 12, 78)( 13, 76)( 14, 75)( 15, 74)( 16, 73)
( 17, 95)( 18, 96)( 19, 93)( 20, 94)( 21, 92)( 22, 91)( 23, 90)( 24, 89)
( 25,111)( 26,112)( 27,109)( 28,110)( 29,108)( 30,107)( 31,106)( 32,105)
( 33,103)( 34,104)( 35,101)( 36,102)( 37,100)( 38, 99)( 39, 98)( 40, 97)
( 41,119)( 42,120)( 43,117)( 44,118)( 45,116)( 46,115)( 47,114)( 48,113)
( 49,135)( 50,136)( 51,133)( 52,134)( 53,132)( 54,131)( 55,130)( 56,129)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,124)( 62,123)( 63,122)( 64,121)
( 65,143)( 66,144)( 67,141)( 68,142)( 69,140)( 70,139)( 71,138)( 72,137);;
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope