Polytope of Type {4,18,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,2}*576
if this polytope has a name.
Group : SmallGroup(576,8262)
Rank : 4
Schlafli Type : {4,18,2}
Number of vertices, edges, etc : 8, 72, 36, 2
Order of s₀s₁s₂s₃ : 18
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,18,2,2} of size 1152
   {4,18,2,3} of size 1728
Vertex Figure Of :
   {2,4,18,2} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,9,2}*288, {4,18,2}*288b, {4,18,2}*288c
   3-fold quotients : {4,6,2}*192
   4-fold quotients : {4,9,2}*144, {2,18,2}*144
   6-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
   8-fold quotients : {2,9,2}*72
   12-fold quotients : {4,3,2}*48, {2,6,2}*48
   24-fold quotients : {2,3,2}*24
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,36,2}*1152b, {4,18,4}*1152b, {4,18,2}*1152b, {4,36,2}*1152c, {8,18,2}*1152b, {8,18,2}*1152c
   3-fold covers : {4,54,2}*1728, {4,18,6}*1728a, {4,18,6}*1728b, {12,18,2}*1728a, {12,18,2}*1728b
Permutation Representation (GAP) :

s0 := (  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)(  8, 43)
(  9, 46)( 10, 45)( 11, 48)( 12, 47)( 13, 50)( 14, 49)( 15, 52)( 16, 51)
( 17, 54)( 18, 53)( 19, 56)( 20, 55)( 21, 58)( 22, 57)( 23, 60)( 24, 59)
( 25, 62)( 26, 61)( 27, 64)( 28, 63)( 29, 66)( 30, 65)( 31, 68)( 32, 67)
( 33, 70)( 34, 69)( 35, 72)( 36, 71)( 73,110)( 74,109)( 75,112)( 76,111)
( 77,114)( 78,113)( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)( 84,119)
( 85,122)( 86,121)( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)( 92,127)
( 93,130)( 94,129)( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)(100,135)
(101,138)(102,137)(103,140)(104,139)(105,142)(106,141)(107,144)(108,143);;
s1 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)( 85,101)
( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)( 93,105)
( 94,107)( 95,106)( 96,108)(110,111)(113,117)(114,119)(115,118)(116,120)
(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)(128,136)
(129,141)(130,143)(131,142)(132,144);;
s2 := (  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)(  8, 95)
(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)( 16, 75)
( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)( 24, 79)
( 25,101)( 26,102)( 27,104)( 28,103)( 29, 97)( 30, 98)( 31,100)( 32, 99)
( 33,105)( 34,106)( 35,108)( 36,107)( 37,121)( 38,122)( 39,124)( 40,123)
( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)( 56,119)
( 57,113)( 58,114)( 59,116)( 60,115)( 61,137)( 62,138)( 63,140)( 64,139)
( 65,133)( 66,134)( 67,136)( 68,135)( 69,141)( 70,142)( 71,144)( 72,143);;
s3 := (145,146);;
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, s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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(146)!(  1, 38)(  2, 37)(  3, 40)(  4, 39)(  5, 42)(  6, 41)(  7, 44)
(  8, 43)(  9, 46)( 10, 45)( 11, 48)( 12, 47)( 13, 50)( 14, 49)( 15, 52)
( 16, 51)( 17, 54)( 18, 53)( 19, 56)( 20, 55)( 21, 58)( 22, 57)( 23, 60)
( 24, 59)( 25, 62)( 26, 61)( 27, 64)( 28, 63)( 29, 66)( 30, 65)( 31, 68)
( 32, 67)( 33, 70)( 34, 69)( 35, 72)( 36, 71)( 73,110)( 74,109)( 75,112)
( 76,111)( 77,114)( 78,113)( 79,116)( 80,115)( 81,118)( 82,117)( 83,120)
( 84,119)( 85,122)( 86,121)( 87,124)( 88,123)( 89,126)( 90,125)( 91,128)
( 92,127)( 93,130)( 94,129)( 95,132)( 96,131)( 97,134)( 98,133)( 99,136)
(100,135)(101,138)(102,137)(103,140)(104,139)(105,142)(106,141)(107,144)
(108,143);
s1 := Sym(146)!(  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)
( 50, 67)( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)
( 58, 71)( 59, 70)( 60, 72)( 74, 75)( 77, 81)( 78, 83)( 79, 82)( 80, 84)
( 85,101)( 86,103)( 87,102)( 88,104)( 89, 97)( 90, 99)( 91, 98)( 92,100)
( 93,105)( 94,107)( 95,106)( 96,108)(110,111)(113,117)(114,119)(115,118)
(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)(127,134)
(128,136)(129,141)(130,143)(131,142)(132,144);
s2 := Sym(146)!(  1, 85)(  2, 86)(  3, 88)(  4, 87)(  5, 93)(  6, 94)(  7, 96)
(  8, 95)(  9, 89)( 10, 90)( 11, 92)( 12, 91)( 13, 73)( 14, 74)( 15, 76)
( 16, 75)( 17, 81)( 18, 82)( 19, 84)( 20, 83)( 21, 77)( 22, 78)( 23, 80)
( 24, 79)( 25,101)( 26,102)( 27,104)( 28,103)( 29, 97)( 30, 98)( 31,100)
( 32, 99)( 33,105)( 34,106)( 35,108)( 36,107)( 37,121)( 38,122)( 39,124)
( 40,123)( 41,129)( 42,130)( 43,132)( 44,131)( 45,125)( 46,126)( 47,128)
( 48,127)( 49,109)( 50,110)( 51,112)( 52,111)( 53,117)( 54,118)( 55,120)
( 56,119)( 57,113)( 58,114)( 59,116)( 60,115)( 61,137)( 62,138)( 63,140)
( 64,139)( 65,133)( 66,134)( 67,136)( 68,135)( 69,141)( 70,142)( 71,144)
( 72,143);
s3 := Sym(146)!(145,146);
poly := sub<Sym(146)|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, 
s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*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 >;

to this polytope