# Polytope of Type {4,18,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,4}*576c
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 : {4,18,2}*288b
3-fold quotients : {4,6,4}*192c
4-fold quotients : {4,9,2}*144
6-fold quotients : {4,6,2}*96c
12-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,36,4}*1152d, {4,36,4}*1152e, {4,18,8}*1152b, {4,18,4}*1152b
3-fold covers : {4,54,4}*1728c, {4,18,12}*1728c, {4,18,12}*1728d
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144);;
s1 := (  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);;
s2 := (  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)( 73,133)( 74,136)( 75,135)( 76,134)( 77,141)( 78,144)
( 79,143)( 80,142)( 81,137)( 82,140)( 83,139)( 84,138)( 85,121)( 86,124)
( 87,123)( 88,122)( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)( 94,128)
( 95,127)( 96,126)( 97,109)( 98,112)( 99,111)(100,110)(101,117)(102,120)
(103,119)(104,118)(105,113)(106,116)(107,115)(108,114);;
s3 := (  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)(  8, 80)
(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144);;
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s1*s0*s1*s2*s0*s1, 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)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144);
s1 := 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);
s2 := 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)( 73,133)( 74,136)( 75,135)( 76,134)( 77,141)
( 78,144)( 79,143)( 80,142)( 81,137)( 82,140)( 83,139)( 84,138)( 85,121)
( 86,124)( 87,123)( 88,122)( 89,129)( 90,132)( 91,131)( 92,130)( 93,125)
( 94,128)( 95,127)( 96,126)( 97,109)( 98,112)( 99,111)(100,110)(101,117)
(102,120)(103,119)(104,118)(105,113)(106,116)(107,115)(108,114);
s3 := Sym(144)!(  1, 73)(  2, 74)(  3, 75)(  4, 76)(  5, 77)(  6, 78)(  7, 79)
(  8, 80)(  9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)
( 32,104)( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)
( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)
( 48,120)( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)
( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)
( 72,144);
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*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s0*s1,
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