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# Polytope of Type {4,18,4}

Atlas Canonical Name : {4,18,4}*576a
Also Known As : {{4,18|2},{18,4|2}}. if this polytope has another name.
Group : SmallGroup(576,1574)
Rank : 4
Schlafli Type : {4,18,4}
Number of vertices, edges, etc : 4, 36, 36, 4
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,18,4,2} of size 1152
Vertex Figure Of :
{2,4,18,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,18,4}*288a, {4,18,2}*288a
3-fold quotients : {4,6,4}*192a
4-fold quotients : {2,18,2}*144
6-fold quotients : {2,6,4}*96a, {4,6,2}*96a
8-fold quotients : {2,9,2}*72
9-fold quotients : {4,2,4}*64
12-fold quotients : {2,6,2}*48
18-fold quotients : {2,2,4}*32, {4,2,2}*32
24-fold quotients : {2,3,2}*24
36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,36,4}*1152a, {4,18,8}*1152a, {8,18,4}*1152a
3-fold covers : {4,54,4}*1728a, {4,18,12}*1728a, {12,18,4}*1728a, {4,18,12}*1728b, {12,18,4}*1728b
Permutation Representation (GAP) :
```s0 := ( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)( 80, 98)
( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)( 88,106)
( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)
(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)(122,140)
(123,141)(124,142)(125,143)(126,144);;
s1 := (  1, 73)(  2, 75)(  3, 74)(  4, 81)(  5, 80)(  6, 79)(  7, 78)(  8, 77)
(  9, 76)( 10, 82)( 11, 84)( 12, 83)( 13, 90)( 14, 89)( 15, 88)( 16, 87)
( 17, 86)( 18, 85)( 19, 91)( 20, 93)( 21, 92)( 22, 99)( 23, 98)( 24, 97)
( 25, 96)( 26, 95)( 27, 94)( 28,100)( 29,102)( 30,101)( 31,108)( 32,107)
( 33,106)( 34,105)( 35,104)( 36,103)( 37,127)( 38,129)( 39,128)( 40,135)
( 41,134)( 42,133)( 43,132)( 44,131)( 45,130)( 46,136)( 47,138)( 48,137)
( 49,144)( 50,143)( 51,142)( 52,141)( 53,140)( 54,139)( 55,109)( 56,111)
( 57,110)( 58,117)( 59,116)( 60,115)( 61,114)( 62,113)( 63,112)( 64,118)
( 65,120)( 66,119)( 67,126)( 68,125)( 69,124)( 70,123)( 71,122)( 72,121);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 13)( 11, 15)( 12, 14)( 16, 18)
( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)( 34, 36)
( 37, 67)( 38, 69)( 39, 68)( 40, 64)( 41, 66)( 42, 65)( 43, 72)( 44, 71)
( 45, 70)( 46, 58)( 47, 60)( 48, 59)( 49, 55)( 50, 57)( 51, 56)( 52, 63)
( 53, 62)( 54, 61)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)
( 84, 86)( 88, 90)( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)
(102,104)(106,108)(109,139)(110,141)(111,140)(112,136)(113,138)(114,137)
(115,144)(116,143)(117,142)(118,130)(119,132)(120,131)(121,127)(122,129)
(123,128)(124,135)(125,134)(126,133);;
s3 := (  1, 37)(  2, 38)(  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)(  8, 44)
(  9, 45)( 10, 46)( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)( 16, 52)
( 17, 53)( 18, 54)( 19, 55)( 20, 56)( 21, 57)( 22, 58)( 23, 59)( 24, 60)
( 25, 61)( 26, 62)( 27, 63)( 28, 64)( 29, 65)( 30, 66)( 31, 67)( 32, 68)
( 33, 69)( 34, 70)( 35, 71)( 36, 72)( 73,127)( 74,128)( 75,129)( 76,130)
( 77,131)( 78,132)( 79,133)( 80,134)( 81,135)( 82,136)( 83,137)( 84,138)
( 85,139)( 86,140)( 87,141)( 88,142)( 89,143)( 90,144)( 91,109)( 92,110)
( 93,111)( 94,112)( 95,113)( 96,114)( 97,115)( 98,116)( 99,117)(100,118)
(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)(108,126);;
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*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, 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)!( 73, 91)( 74, 92)( 75, 93)( 76, 94)( 77, 95)( 78, 96)( 79, 97)
( 80, 98)( 81, 99)( 82,100)( 83,101)( 84,102)( 85,103)( 86,104)( 87,105)
( 88,106)( 89,107)( 90,108)(109,127)(110,128)(111,129)(112,130)(113,131)
(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)(121,139)
(122,140)(123,141)(124,142)(125,143)(126,144);
s1 := Sym(144)!(  1, 73)(  2, 75)(  3, 74)(  4, 81)(  5, 80)(  6, 79)(  7, 78)
(  8, 77)(  9, 76)( 10, 82)( 11, 84)( 12, 83)( 13, 90)( 14, 89)( 15, 88)
( 16, 87)( 17, 86)( 18, 85)( 19, 91)( 20, 93)( 21, 92)( 22, 99)( 23, 98)
( 24, 97)( 25, 96)( 26, 95)( 27, 94)( 28,100)( 29,102)( 30,101)( 31,108)
( 32,107)( 33,106)( 34,105)( 35,104)( 36,103)( 37,127)( 38,129)( 39,128)
( 40,135)( 41,134)( 42,133)( 43,132)( 44,131)( 45,130)( 46,136)( 47,138)
( 48,137)( 49,144)( 50,143)( 51,142)( 52,141)( 53,140)( 54,139)( 55,109)
( 56,111)( 57,110)( 58,117)( 59,116)( 60,115)( 61,114)( 62,113)( 63,112)
( 64,118)( 65,120)( 66,119)( 67,126)( 68,125)( 69,124)( 70,123)( 71,122)
( 72,121);
s2 := Sym(144)!(  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 13)( 11, 15)( 12, 14)
( 16, 18)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)
( 34, 36)( 37, 67)( 38, 69)( 39, 68)( 40, 64)( 41, 66)( 42, 65)( 43, 72)
( 44, 71)( 45, 70)( 46, 58)( 47, 60)( 48, 59)( 49, 55)( 50, 57)( 51, 56)
( 52, 63)( 53, 62)( 54, 61)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)
( 83, 87)( 84, 86)( 88, 90)( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)
(101,105)(102,104)(106,108)(109,139)(110,141)(111,140)(112,136)(113,138)
(114,137)(115,144)(116,143)(117,142)(118,130)(119,132)(120,131)(121,127)
(122,129)(123,128)(124,135)(125,134)(126,133);
s3 := Sym(144)!(  1, 37)(  2, 38)(  3, 39)(  4, 40)(  5, 41)(  6, 42)(  7, 43)
(  8, 44)(  9, 45)( 10, 46)( 11, 47)( 12, 48)( 13, 49)( 14, 50)( 15, 51)
( 16, 52)( 17, 53)( 18, 54)( 19, 55)( 20, 56)( 21, 57)( 22, 58)( 23, 59)
( 24, 60)( 25, 61)( 26, 62)( 27, 63)( 28, 64)( 29, 65)( 30, 66)( 31, 67)
( 32, 68)( 33, 69)( 34, 70)( 35, 71)( 36, 72)( 73,127)( 74,128)( 75,129)
( 76,130)( 77,131)( 78,132)( 79,133)( 80,134)( 81,135)( 82,136)( 83,137)
( 84,138)( 85,139)( 86,140)( 87,141)( 88,142)( 89,143)( 90,144)( 91,109)
( 92,110)( 93,111)( 94,112)( 95,113)( 96,114)( 97,115)( 98,116)( 99,117)
(100,118)(101,119)(102,120)(103,121)(104,122)(105,123)(106,124)(107,125)
(108,126);
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
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