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Polytope of Type {8,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,18}*576c
if this polytope has a name.
Group : SmallGroup(576,4983)
Rank : 3
Schlafli Type : {8,18}
Number of vertices, edges, etc : 16, 144, 36
Order of s0s1s2 : 18
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,18,2} of size 1152
Vertex Figure Of :
   {2,8,18} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,18}*288
   3-fold quotients : {8,6}*192c
   4-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
   6-fold quotients : {4,6}*96
   8-fold quotients : {4,9}*72, {2,18}*72
   12-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   16-fold quotients : {2,9}*36
   24-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {2,3}*12
   72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,36}*1152f, {8,18}*1152f, {8,36}*1152g
   3-fold covers : {8,54}*1728c, {24,18}*1728c, {24,18}*1728d
Permutation Representation (GAP) :
s0 := (  1, 77)(  2, 78)(  3, 80)(  4, 79)(  5, 74)(  6, 73)(  7, 75)(  8, 76)
(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 82)( 14, 81)( 15, 83)( 16, 84)
( 17, 93)( 18, 94)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 91)( 24, 92)
( 25,101)( 26,102)( 27,104)( 28,103)( 29, 98)( 30, 97)( 31, 99)( 32,100)
( 33,109)( 34,110)( 35,112)( 36,111)( 37,106)( 38,105)( 39,107)( 40,108)
( 41,117)( 42,118)( 43,120)( 44,119)( 45,114)( 46,113)( 47,115)( 48,116)
( 49,125)( 50,126)( 51,128)( 52,127)( 53,122)( 54,121)( 55,123)( 56,124)
( 57,133)( 58,134)( 59,136)( 60,135)( 61,130)( 62,129)( 63,131)( 64,132)
( 65,141)( 66,142)( 67,144)( 68,143)( 69,138)( 70,137)( 71,139)( 72,140);;
s1 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)
( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)
( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)
( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 75, 78)( 76, 77)( 79, 80)( 81, 89)( 82, 90)
( 83, 94)( 84, 93)( 85, 92)( 86, 91)( 87, 96)( 88, 95)( 97,137)( 98,138)
( 99,142)(100,141)(101,140)(102,139)(103,144)(104,143)(105,129)(106,130)
(107,134)(108,133)(109,132)(110,131)(111,136)(112,135)(113,121)(114,122)
(115,126)(116,125)(117,124)(118,123)(119,128)(120,127);;
s2 := (  1, 49)(  2, 50)(  3, 55)(  4, 56)(  5, 54)(  6, 53)(  7, 51)(  8, 52)
(  9, 65)( 10, 66)( 11, 71)( 12, 72)( 13, 70)( 14, 69)( 15, 67)( 16, 68)
( 17, 57)( 18, 58)( 19, 63)( 20, 64)( 21, 62)( 22, 61)( 23, 59)( 24, 60)
( 27, 31)( 28, 32)( 29, 30)( 33, 41)( 34, 42)( 35, 47)( 36, 48)( 37, 46)
( 38, 45)( 39, 43)( 40, 44)( 73,122)( 74,121)( 75,128)( 76,127)( 77,125)
( 78,126)( 79,124)( 80,123)( 81,138)( 82,137)( 83,144)( 84,143)( 85,141)
( 86,142)( 87,140)( 88,139)( 89,130)( 90,129)( 91,136)( 92,135)( 93,133)
( 94,134)( 95,132)( 96,131)( 97, 98)( 99,104)(100,103)(105,114)(106,113)
(107,120)(108,119)(109,117)(110,118)(111,116)(112,115);;
poly := Group([s0,s1,s2]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*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, 77)(  2, 78)(  3, 80)(  4, 79)(  5, 74)(  6, 73)(  7, 75)
(  8, 76)(  9, 85)( 10, 86)( 11, 88)( 12, 87)( 13, 82)( 14, 81)( 15, 83)
( 16, 84)( 17, 93)( 18, 94)( 19, 96)( 20, 95)( 21, 90)( 22, 89)( 23, 91)
( 24, 92)( 25,101)( 26,102)( 27,104)( 28,103)( 29, 98)( 30, 97)( 31, 99)
( 32,100)( 33,109)( 34,110)( 35,112)( 36,111)( 37,106)( 38,105)( 39,107)
( 40,108)( 41,117)( 42,118)( 43,120)( 44,119)( 45,114)( 46,113)( 47,115)
( 48,116)( 49,125)( 50,126)( 51,128)( 52,127)( 53,122)( 54,121)( 55,123)
( 56,124)( 57,133)( 58,134)( 59,136)( 60,135)( 61,130)( 62,129)( 63,131)
( 64,132)( 65,141)( 66,142)( 67,144)( 68,143)( 69,138)( 70,137)( 71,139)
( 72,140);
s1 := Sym(144)!(  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)
( 13, 20)( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)
( 29, 68)( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)
( 37, 60)( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 75, 78)( 76, 77)( 79, 80)( 81, 89)
( 82, 90)( 83, 94)( 84, 93)( 85, 92)( 86, 91)( 87, 96)( 88, 95)( 97,137)
( 98,138)( 99,142)(100,141)(101,140)(102,139)(103,144)(104,143)(105,129)
(106,130)(107,134)(108,133)(109,132)(110,131)(111,136)(112,135)(113,121)
(114,122)(115,126)(116,125)(117,124)(118,123)(119,128)(120,127);
s2 := Sym(144)!(  1, 49)(  2, 50)(  3, 55)(  4, 56)(  5, 54)(  6, 53)(  7, 51)
(  8, 52)(  9, 65)( 10, 66)( 11, 71)( 12, 72)( 13, 70)( 14, 69)( 15, 67)
( 16, 68)( 17, 57)( 18, 58)( 19, 63)( 20, 64)( 21, 62)( 22, 61)( 23, 59)
( 24, 60)( 27, 31)( 28, 32)( 29, 30)( 33, 41)( 34, 42)( 35, 47)( 36, 48)
( 37, 46)( 38, 45)( 39, 43)( 40, 44)( 73,122)( 74,121)( 75,128)( 76,127)
( 77,125)( 78,126)( 79,124)( 80,123)( 81,138)( 82,137)( 83,144)( 84,143)
( 85,141)( 86,142)( 87,140)( 88,139)( 89,130)( 90,129)( 91,136)( 92,135)
( 93,133)( 94,134)( 95,132)( 96,131)( 97, 98)( 99,104)(100,103)(105,114)
(106,113)(107,120)(108,119)(109,117)(110,118)(111,116)(112,115);
poly := sub<Sym(144)|s0,s1,s2>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*s0*s1*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.
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