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# Polytope of Type {4,4,9}

Atlas Canonical Name : {4,4,9}*576b
if this polytope has a name.
Group : SmallGroup(576,4970)
Rank : 4
Schlafli Type : {4,4,9}
Number of vertices, edges, etc : 4, 16, 36, 18
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,4,9,2} of size 1152
Vertex Figure Of :
{2,4,4,9} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,4,9}*288
3-fold quotients : {4,4,3}*192b
4-fold quotients : {4,2,9}*144, {2,4,9}*144
6-fold quotients : {2,4,3}*96
8-fold quotients : {2,2,9}*72
12-fold quotients : {4,2,3}*48, {2,4,3}*48
24-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,8,9}*1152, {8,4,9}*1152, {4,4,18}*1152d
3-fold covers : {4,4,27}*1728b, {12,4,9}*1728, {4,12,9}*1728
Permutation Representation (GAP) :
```s0 := ( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)( 80,116)
( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)( 88,124)
( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)( 96,132)
( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)(104,140)
(105,141)(106,142)(107,143)(108,144);;
s1 := (  1, 75)(  2, 76)(  3, 73)(  4, 74)(  5, 79)(  6, 80)(  7, 77)(  8, 78)
(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 87)( 14, 88)( 15, 85)( 16, 86)
( 17, 91)( 18, 92)( 19, 89)( 20, 90)( 21, 95)( 22, 96)( 23, 93)( 24, 94)
( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)( 32,102)
( 33,107)( 34,108)( 35,105)( 36,106)( 37,111)( 38,112)( 39,109)( 40,110)
( 41,115)( 42,116)( 43,113)( 44,114)( 45,119)( 46,120)( 47,117)( 48,118)
( 49,123)( 50,124)( 51,121)( 52,122)( 53,127)( 54,128)( 55,125)( 56,126)
( 57,131)( 58,132)( 59,129)( 60,130)( 61,135)( 62,136)( 63,133)( 64,134)
( 65,139)( 66,140)( 67,137)( 68,138)( 69,143)( 70,144)( 71,141)( 72,142);;
s2 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 33)( 14, 35)( 15, 34)
( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)( 23, 26)
( 24, 28)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 69)( 50, 71)
( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)( 58, 63)
( 59, 62)( 60, 64)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85,105)
( 86,107)( 87,106)( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)( 93, 97)
( 94, 99)( 95, 98)( 96,100)(110,111)(113,117)(114,119)(115,118)(116,120)
(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)(127,138)(128,140)
(129,133)(130,135)(131,134)(132,136);;
s3 := (  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)(  8, 34)
(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)( 19, 23)
( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)( 43, 71)
( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)( 54, 60)
( 55, 59)( 56, 58)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,105)( 78,108)
( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)( 84,102)( 86, 88)( 89, 93)
( 90, 96)( 91, 95)( 92, 94)(109,133)(110,136)(111,135)(112,134)(113,141)
(114,144)(115,143)(116,142)(117,137)(118,140)(119,139)(120,138)(122,124)
(125,129)(126,132)(127,131)(128,130);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(144)!( 73,109)( 74,110)( 75,111)( 76,112)( 77,113)( 78,114)( 79,115)
( 80,116)( 81,117)( 82,118)( 83,119)( 84,120)( 85,121)( 86,122)( 87,123)
( 88,124)( 89,125)( 90,126)( 91,127)( 92,128)( 93,129)( 94,130)( 95,131)
( 96,132)( 97,133)( 98,134)( 99,135)(100,136)(101,137)(102,138)(103,139)
(104,140)(105,141)(106,142)(107,143)(108,144);
s1 := Sym(144)!(  1, 75)(  2, 76)(  3, 73)(  4, 74)(  5, 79)(  6, 80)(  7, 77)
(  8, 78)(  9, 83)( 10, 84)( 11, 81)( 12, 82)( 13, 87)( 14, 88)( 15, 85)
( 16, 86)( 17, 91)( 18, 92)( 19, 89)( 20, 90)( 21, 95)( 22, 96)( 23, 93)
( 24, 94)( 25, 99)( 26,100)( 27, 97)( 28, 98)( 29,103)( 30,104)( 31,101)
( 32,102)( 33,107)( 34,108)( 35,105)( 36,106)( 37,111)( 38,112)( 39,109)
( 40,110)( 41,115)( 42,116)( 43,113)( 44,114)( 45,119)( 46,120)( 47,117)
( 48,118)( 49,123)( 50,124)( 51,121)( 52,122)( 53,127)( 54,128)( 55,125)
( 56,126)( 57,131)( 58,132)( 59,129)( 60,130)( 61,135)( 62,136)( 63,133)
( 64,134)( 65,139)( 66,140)( 67,137)( 68,138)( 69,143)( 70,144)( 71,141)
( 72,142);
s2 := Sym(144)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 33)( 14, 35)
( 15, 34)( 16, 36)( 17, 29)( 18, 31)( 19, 30)( 20, 32)( 21, 25)( 22, 27)
( 23, 26)( 24, 28)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 69)
( 50, 71)( 51, 70)( 52, 72)( 53, 65)( 54, 67)( 55, 66)( 56, 68)( 57, 61)
( 58, 63)( 59, 62)( 60, 64)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)
( 85,105)( 86,107)( 87,106)( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)
( 93, 97)( 94, 99)( 95, 98)( 96,100)(110,111)(113,117)(114,119)(115,118)
(116,120)(121,141)(122,143)(123,142)(124,144)(125,137)(126,139)(127,138)
(128,140)(129,133)(130,135)(131,134)(132,136);
s3 := Sym(144)!(  1, 25)(  2, 28)(  3, 27)(  4, 26)(  5, 33)(  6, 36)(  7, 35)
(  8, 34)(  9, 29)( 10, 32)( 11, 31)( 12, 30)( 14, 16)( 17, 21)( 18, 24)
( 19, 23)( 20, 22)( 37, 61)( 38, 64)( 39, 63)( 40, 62)( 41, 69)( 42, 72)
( 43, 71)( 44, 70)( 45, 65)( 46, 68)( 47, 67)( 48, 66)( 50, 52)( 53, 57)
( 54, 60)( 55, 59)( 56, 58)( 73, 97)( 74,100)( 75, 99)( 76, 98)( 77,105)
( 78,108)( 79,107)( 80,106)( 81,101)( 82,104)( 83,103)( 84,102)( 86, 88)
( 89, 93)( 90, 96)( 91, 95)( 92, 94)(109,133)(110,136)(111,135)(112,134)
(113,141)(114,144)(115,143)(116,142)(117,137)(118,140)(119,139)(120,138)
(122,124)(125,129)(126,132)(127,131)(128,130);
poly := sub<Sym(144)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope