Polytope of Type {18,8}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,8}*576a
if this polytope has a name.
Group : SmallGroup(576,1381)
Rank : 3
Schlafli Type : {18,8}
Number of vertices, edges, etc : 36, 144, 16
Order of s0s1s2 : 9
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {18,8,2} of size 1152
Vertex Figure Of :
   {2,18,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,8}*192a
   4-fold quotients : {18,4}*144c
   8-fold quotients : {9,4}*72
   12-fold quotients : {6,4}*48b
   24-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {36,8}*1152c, {36,8}*1152d, {18,8}*1152d
   3-fold covers : {54,8}*1728a, {18,24}*1728a
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2> of order 2.
      8 facets:
         8 of {18}*36
      24 vertex figures:
         12 of {8}*16
         12 of {4}*8

Permutation Representation (GAP) : 
s0 := (  3,  4)(  5, 10)(  6,  9)(  7, 11)(  8, 12)( 15, 16)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 42)( 22, 41)( 23, 43)( 24, 44)( 25, 38)( 26, 37)( 27, 39)( 28, 40)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)( 50,114)( 51,116)( 52,115)( 53,122)( 54,121)( 55,123)( 56,124)( 57,118)( 58,117)( 59,119)( 60,120)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,106)( 70,105)( 71,107)( 72,108)( 73,102)( 74,101)( 75,103)( 76,104)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)( 82,130)( 83,132)( 84,131)( 85,138)( 86,137)( 87,139)( 88,140)( 89,134)( 90,133)( 91,135)( 92,136)( 93,141)( 94,142)( 95,144)( 96,143);;
s1 := (  1, 49)(  2, 51)(  3, 50)(  4, 52)(  5, 56)(  6, 54)(  7, 55)(  8, 53)(  9, 64)( 10, 62)( 11, 63)( 12, 61)( 13, 60)( 14, 58)( 15, 59)( 16, 57)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 88)( 22, 86)( 23, 87)( 24, 85)( 25, 96)( 26, 94)( 27, 95)( 28, 93)( 29, 92)( 30, 90)( 31, 91)( 32, 89)( 33, 65)( 34, 67)( 35, 66)( 36, 68)( 37, 72)( 38, 70)( 39, 71)( 40, 69)( 41, 80)( 42, 78)( 43, 79)( 44, 77)( 45, 76)( 46, 74)( 47, 75)( 48, 73)( 97,113)( 98,115)( 99,114)(100,116)(101,120)(102,118)(103,119)(104,117)(105,128)(106,126)(107,127)(108,125)(109,124)(110,122)(111,123)(112,121)(130,131)(133,136)(137,144)(138,142)(139,143)(140,141);;
s2 := (  1, 13)(  2, 14)(  3, 15)(  4, 16)(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)(104,108)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124)(129,141)(130,142)(131,143)(132,144)(133,137)(134,138)(135,139)(136,140);;
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(144)!(  3,  4)(  5, 10)(  6,  9)(  7, 11)(  8, 12)( 15, 16)( 17, 33)( 18, 34)( 19, 36)( 20, 35)( 21, 42)( 22, 41)( 23, 43)( 24, 44)( 25, 38)( 26, 37)( 27, 39)( 28, 40)( 29, 45)( 30, 46)( 31, 48)( 32, 47)( 49,113)( 50,114)( 51,116)( 52,115)( 53,122)( 54,121)( 55,123)( 56,124)( 57,118)( 58,117)( 59,119)( 60,120)( 61,125)( 62,126)( 63,128)( 64,127)( 65, 97)( 66, 98)( 67,100)( 68, 99)( 69,106)( 70,105)( 71,107)( 72,108)( 73,102)( 74,101)( 75,103)( 76,104)( 77,109)( 78,110)( 79,112)( 80,111)( 81,129)( 82,130)( 83,132)( 84,131)( 85,138)( 86,137)( 87,139)( 88,140)( 89,134)( 90,133)( 91,135)( 92,136)( 93,141)( 94,142)( 95,144)( 96,143);
s1 := Sym(144)!(  1, 49)(  2, 51)(  3, 50)(  4, 52)(  5, 56)(  6, 54)(  7, 55)(  8, 53)(  9, 64)( 10, 62)( 11, 63)( 12, 61)( 13, 60)( 14, 58)( 15, 59)( 16, 57)( 17, 81)( 18, 83)( 19, 82)( 20, 84)( 21, 88)( 22, 86)( 23, 87)( 24, 85)( 25, 96)( 26, 94)( 27, 95)( 28, 93)( 29, 92)( 30, 90)( 31, 91)( 32, 89)( 33, 65)( 34, 67)( 35, 66)( 36, 68)( 37, 72)( 38, 70)( 39, 71)( 40, 69)( 41, 80)( 42, 78)( 43, 79)( 44, 77)( 45, 76)( 46, 74)( 47, 75)( 48, 73)( 97,113)( 98,115)( 99,114)(100,116)(101,120)(102,118)(103,119)(104,117)(105,128)(106,126)(107,127)(108,125)(109,124)(110,122)(111,123)(112,121)(130,131)(133,136)(137,144)(138,142)(139,143)(140,141);
s2 := Sym(144)!(  1, 13)(  2, 14)(  3, 15)(  4, 16)(  5,  9)(  6, 10)(  7, 11)(  8, 12)( 17, 29)( 18, 30)( 19, 31)( 20, 32)( 21, 25)( 22, 26)( 23, 27)( 24, 28)( 33, 45)( 34, 46)( 35, 47)( 36, 48)( 37, 41)( 38, 42)( 39, 43)( 40, 44)( 49, 61)( 50, 62)( 51, 63)( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 77)( 66, 78)( 67, 79)( 68, 80)( 69, 73)( 70, 74)( 71, 75)( 72, 76)( 81, 93)( 82, 94)( 83, 95)( 84, 96)( 85, 89)( 86, 90)( 87, 91)( 88, 92)( 97,109)( 98,110)( 99,111)(100,112)(101,105)(102,106)(103,107)(104,108)(113,125)(114,126)(115,127)(116,128)(117,121)(118,122)(119,123)(120,124)(129,141)(130,142)(131,143)(132,144)(133,137)(134,138)(135,139)(136,140);
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*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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