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# Polytope of Type {4,18,4}

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