Polytope of Type {8,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6}*288
if this polytope has a name.
Group : SmallGroup(288,873)
Rank : 3
Schlafli Type : {8,6}
Number of vertices, edges, etc : 24, 72, 18
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {8,6,2} of size 576
   {8,6,3} of size 864
   {8,6,4} of size 1152
   {8,6,6} of size 1728
   {8,6,6} of size 1728
Vertex Figure Of :
   {2,8,6} of size 576
   {4,8,6} of size 1152
   {4,8,6} of size 1152
   {6,8,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,6}*144
   4-fold quotients : {4,6}*72
   9-fold quotients : {8,2}*32
   18-fold quotients : {4,2}*16
   36-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {16,6}*576, {8,12}*576a
   3-fold covers : {8,6}*864a, {24,6}*864d, {24,6}*864e, {8,6}*864b, {24,6}*864g, {24,6}*864h
   4-fold covers : {8,12}*1152a, {8,24}*1152b, {8,24}*1152c, {16,12}*1152a, {16,12}*1152b, {32,6}*1152
   5-fold covers : {8,30}*1440, {40,6}*1440
   6-fold covers : {16,6}*1728a, {48,6}*1728d, {48,6}*1728e, {8,12}*1728a, {24,12}*1728g, {24,12}*1728h, {16,6}*1728b, {48,6}*1728g, {8,12}*1728g, {24,12}*1728s, {48,6}*1728h, {24,12}*1728u
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {8}*16
      8 vertex figures:
         8 of {6}*12
   P/N, where N=<s0*s1*s2*s1*s0*s2> of order 3.
      6 facets:
         6 of {8}*16
      16 vertex figures:
         4 of {6}*12
         12 of {2}*4

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