Polytope of Type {2,18,4}

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

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

to this polytope