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# Polytope of Type {4,4,9}

Atlas Canonical Name : {4,4,9}*576a
if this polytope has a name.
Group : SmallGroup(576,1380)
Rank : 4
Schlafli Type : {4,4,9}
Number of vertices, edges, etc : 8, 16, 36, 9
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Non-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
{3,4,4,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {4,4,3}*192a
4-fold quotients : {2,4,9}*144
12-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,4,9}*1152a, {4,4,9}*1152b, {4,4,18}*1152b, {4,4,18}*1152c
3-fold covers : {4,4,27}*1728a
Permutation Representation (GAP) :
```s0 := (  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 21, 23)( 22, 24)
( 25, 28)( 26, 27)( 29, 30)( 31, 32)( 37, 39)( 38, 40)( 41, 44)( 42, 43)
( 45, 46)( 47, 48)( 53, 55)( 54, 56)( 57, 60)( 58, 59)( 61, 62)( 63, 64)
( 69, 71)( 70, 72)( 73, 76)( 74, 75)( 77, 78)( 79, 80)( 85, 87)( 86, 88)
( 89, 92)( 90, 91)( 93, 94)( 95, 96)(101,103)(102,104)(105,108)(106,107)
(109,110)(111,112)(117,119)(118,120)(121,124)(122,123)(125,126)(127,128)
(133,135)(134,136)(137,140)(138,139)(141,142)(143,144);;
s1 := (  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)(  8, 16)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)( 24, 32)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)( 40, 48)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)( 56, 64)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)( 72, 80)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)( 88, 96)
( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)(104,112)
(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)(120,128)
(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)(136,144);;
s2 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 37)( 26, 38)
( 27, 40)( 28, 39)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)( 50,114)
( 51,116)( 52,115)( 53,121)( 54,122)( 55,124)( 56,123)( 57,117)( 58,118)
( 59,120)( 60,119)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)( 66, 98)
( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)( 73,101)( 74,102)
( 75,104)( 76,103)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)( 82,130)
( 83,132)( 84,131)( 85,137)( 86,138)( 87,140)( 88,139)( 89,133)( 90,134)
( 91,136)( 92,135)( 93,141)( 94,142)( 95,144)( 96,143);;
s3 := (  1, 97)(  2, 99)(  3, 98)(  4,100)(  5,109)(  6,111)(  7,110)(  8,112)
(  9,105)( 10,107)( 11,106)( 12,108)( 13,101)( 14,103)( 15,102)( 16,104)
( 17,129)( 18,131)( 19,130)( 20,132)( 21,141)( 22,143)( 23,142)( 24,144)
( 25,137)( 26,139)( 27,138)( 28,140)( 29,133)( 30,135)( 31,134)( 32,136)
( 33,113)( 34,115)( 35,114)( 36,116)( 37,125)( 38,127)( 39,126)( 40,128)
( 41,121)( 42,123)( 43,122)( 44,124)( 45,117)( 46,119)( 47,118)( 48,120)
( 50, 51)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 58, 59)( 65, 81)( 66, 83)
( 67, 82)( 68, 84)( 69, 93)( 70, 95)( 71, 94)( 72, 96)( 73, 89)( 74, 91)
( 75, 90)( 76, 92)( 77, 85)( 78, 87)( 79, 86)( 80, 88);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
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)!(  5,  7)(  6,  8)(  9, 12)( 10, 11)( 13, 14)( 15, 16)( 21, 23)
( 22, 24)( 25, 28)( 26, 27)( 29, 30)( 31, 32)( 37, 39)( 38, 40)( 41, 44)
( 42, 43)( 45, 46)( 47, 48)( 53, 55)( 54, 56)( 57, 60)( 58, 59)( 61, 62)
( 63, 64)( 69, 71)( 70, 72)( 73, 76)( 74, 75)( 77, 78)( 79, 80)( 85, 87)
( 86, 88)( 89, 92)( 90, 91)( 93, 94)( 95, 96)(101,103)(102,104)(105,108)
(106,107)(109,110)(111,112)(117,119)(118,120)(121,124)(122,123)(125,126)
(127,128)(133,135)(134,136)(137,140)(138,139)(141,142)(143,144);
s1 := Sym(144)!(  1,  9)(  2, 10)(  3, 11)(  4, 12)(  5, 13)(  6, 14)(  7, 15)
(  8, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 29)( 22, 30)( 23, 31)
( 24, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 45)( 38, 46)( 39, 47)
( 40, 48)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 61)( 54, 62)( 55, 63)
( 56, 64)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 77)( 70, 78)( 71, 79)
( 72, 80)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 93)( 86, 94)( 87, 95)
( 88, 96)( 97,105)( 98,106)( 99,107)(100,108)(101,109)(102,110)(103,111)
(104,112)(113,121)(114,122)(115,123)(116,124)(117,125)(118,126)(119,127)
(120,128)(129,137)(130,138)(131,139)(132,140)(133,141)(134,142)(135,143)
(136,144);
s2 := Sym(144)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 15, 16)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 41)( 22, 42)( 23, 44)( 24, 43)( 25, 37)
( 26, 38)( 27, 40)( 28, 39)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)
( 50,114)( 51,116)( 52,115)( 53,121)( 54,122)( 55,124)( 56,123)( 57,117)
( 58,118)( 59,120)( 60,119)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)
( 66, 98)( 67,100)( 68, 99)( 69,105)( 70,106)( 71,108)( 72,107)( 73,101)
( 74,102)( 75,104)( 76,103)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)
( 82,130)( 83,132)( 84,131)( 85,137)( 86,138)( 87,140)( 88,139)( 89,133)
( 90,134)( 91,136)( 92,135)( 93,141)( 94,142)( 95,144)( 96,143);
s3 := Sym(144)!(  1, 97)(  2, 99)(  3, 98)(  4,100)(  5,109)(  6,111)(  7,110)
(  8,112)(  9,105)( 10,107)( 11,106)( 12,108)( 13,101)( 14,103)( 15,102)
( 16,104)( 17,129)( 18,131)( 19,130)( 20,132)( 21,141)( 22,143)( 23,142)
( 24,144)( 25,137)( 26,139)( 27,138)( 28,140)( 29,133)( 30,135)( 31,134)
( 32,136)( 33,113)( 34,115)( 35,114)( 36,116)( 37,125)( 38,127)( 39,126)
( 40,128)( 41,121)( 42,123)( 43,122)( 44,124)( 45,117)( 46,119)( 47,118)
( 48,120)( 50, 51)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 58, 59)( 65, 81)
( 66, 83)( 67, 82)( 68, 84)( 69, 93)( 70, 95)( 71, 94)( 72, 96)( 73, 89)
( 74, 91)( 75, 90)( 76, 92)( 77, 85)( 78, 87)( 79, 86)( 80, 88);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope