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Polytope of Type {2,8,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,8,6}*576
if this polytope has a name.
Group : SmallGroup(576,8380)
Rank : 4
Schlafli Type : {2,8,6}
Number of vertices, edges, etc : 2, 24, 72, 18
Order of s0s1s2s3 : 8
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,8,6,2} of size 1152
   {2,8,6,3} of size 1728
Vertex Figure Of :
   {2,2,8,6} of size 1152
   {3,2,8,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,6}*288
   4-fold quotients : {2,4,6}*144
   9-fold quotients : {2,8,2}*64
   18-fold quotients : {2,4,2}*32
   36-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,8,12}*1152a, {4,8,6}*1152a, {2,16,6}*1152
   3-fold covers : {2,8,6}*1728a, {2,24,6}*1728d, {2,24,6}*1728e, {6,8,6}*1728b, {2,8,6}*1728b, {2,24,6}*1728g, {2,24,6}*1728h
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 21, 30)( 22, 31)
( 23, 32)( 24, 36)( 25, 37)( 26, 38)( 27, 33)( 28, 34)( 29, 35)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)( 75, 93)( 76, 94)
( 77, 95)( 78, 99)( 79,100)( 80,101)( 81, 96)( 82, 97)( 83, 98)( 84,102)
( 85,103)( 86,104)( 87,108)( 88,109)( 89,110)( 90,105)( 91,106)( 92,107)
(111,138)(112,139)(113,140)(114,144)(115,145)(116,146)(117,141)(118,142)
(119,143)(120,129)(121,130)(122,131)(123,135)(124,136)(125,137)(126,132)
(127,133)(128,134);;
s2 := (  3, 75)(  4, 78)(  5, 81)(  6, 76)(  7, 79)(  8, 82)(  9, 77)( 10, 80)
( 11, 83)( 12, 84)( 13, 87)( 14, 90)( 15, 85)( 16, 88)( 17, 91)( 18, 86)
( 19, 89)( 20, 92)( 21,102)( 22,105)( 23,108)( 24,103)( 25,106)( 26,109)
( 27,104)( 28,107)( 29,110)( 30, 93)( 31, 96)( 32, 99)( 33, 94)( 34, 97)
( 35,100)( 36, 95)( 37, 98)( 38,101)( 39,120)( 40,123)( 41,126)( 42,121)
( 43,124)( 44,127)( 45,122)( 46,125)( 47,128)( 48,111)( 49,114)( 50,117)
( 51,112)( 52,115)( 53,118)( 54,113)( 55,116)( 56,119)( 57,129)( 58,132)
( 59,135)( 60,130)( 61,133)( 62,136)( 63,131)( 64,134)( 65,137)( 66,138)
( 67,141)( 68,144)( 69,139)( 70,142)( 71,145)( 72,140)( 73,143)( 74,146);;
s3 := (  3, 58)(  4, 57)(  5, 59)(  6, 64)(  7, 63)(  8, 65)(  9, 61)( 10, 60)
( 11, 62)( 12, 67)( 13, 66)( 14, 68)( 15, 73)( 16, 72)( 17, 74)( 18, 70)
( 19, 69)( 20, 71)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)( 26, 56)
( 27, 52)( 28, 51)( 29, 53)( 30, 40)( 31, 39)( 32, 41)( 33, 46)( 34, 45)
( 35, 47)( 36, 43)( 37, 42)( 38, 44)( 75,130)( 76,129)( 77,131)( 78,136)
( 79,135)( 80,137)( 81,133)( 82,132)( 83,134)( 84,139)( 85,138)( 86,140)
( 87,145)( 88,144)( 89,146)( 90,142)( 91,141)( 92,143)( 93,121)( 94,120)
( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)(100,123)(101,125)(102,112)
(103,111)(104,113)(105,118)(106,117)(107,119)(108,115)(109,114)(110,116);;
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*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*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)!(  6,  9)(  7, 10)(  8, 11)( 15, 18)( 16, 19)( 17, 20)( 21, 30)
( 22, 31)( 23, 32)( 24, 36)( 25, 37)( 26, 38)( 27, 33)( 28, 34)( 29, 35)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 69, 72)( 70, 73)( 71, 74)( 75, 93)
( 76, 94)( 77, 95)( 78, 99)( 79,100)( 80,101)( 81, 96)( 82, 97)( 83, 98)
( 84,102)( 85,103)( 86,104)( 87,108)( 88,109)( 89,110)( 90,105)( 91,106)
( 92,107)(111,138)(112,139)(113,140)(114,144)(115,145)(116,146)(117,141)
(118,142)(119,143)(120,129)(121,130)(122,131)(123,135)(124,136)(125,137)
(126,132)(127,133)(128,134);
s2 := Sym(146)!(  3, 75)(  4, 78)(  5, 81)(  6, 76)(  7, 79)(  8, 82)(  9, 77)
( 10, 80)( 11, 83)( 12, 84)( 13, 87)( 14, 90)( 15, 85)( 16, 88)( 17, 91)
( 18, 86)( 19, 89)( 20, 92)( 21,102)( 22,105)( 23,108)( 24,103)( 25,106)
( 26,109)( 27,104)( 28,107)( 29,110)( 30, 93)( 31, 96)( 32, 99)( 33, 94)
( 34, 97)( 35,100)( 36, 95)( 37, 98)( 38,101)( 39,120)( 40,123)( 41,126)
( 42,121)( 43,124)( 44,127)( 45,122)( 46,125)( 47,128)( 48,111)( 49,114)
( 50,117)( 51,112)( 52,115)( 53,118)( 54,113)( 55,116)( 56,119)( 57,129)
( 58,132)( 59,135)( 60,130)( 61,133)( 62,136)( 63,131)( 64,134)( 65,137)
( 66,138)( 67,141)( 68,144)( 69,139)( 70,142)( 71,145)( 72,140)( 73,143)
( 74,146);
s3 := Sym(146)!(  3, 58)(  4, 57)(  5, 59)(  6, 64)(  7, 63)(  8, 65)(  9, 61)
( 10, 60)( 11, 62)( 12, 67)( 13, 66)( 14, 68)( 15, 73)( 16, 72)( 17, 74)
( 18, 70)( 19, 69)( 20, 71)( 21, 49)( 22, 48)( 23, 50)( 24, 55)( 25, 54)
( 26, 56)( 27, 52)( 28, 51)( 29, 53)( 30, 40)( 31, 39)( 32, 41)( 33, 46)
( 34, 45)( 35, 47)( 36, 43)( 37, 42)( 38, 44)( 75,130)( 76,129)( 77,131)
( 78,136)( 79,135)( 80,137)( 81,133)( 82,132)( 83,134)( 84,139)( 85,138)
( 86,140)( 87,145)( 88,144)( 89,146)( 90,142)( 91,141)( 92,143)( 93,121)
( 94,120)( 95,122)( 96,127)( 97,126)( 98,128)( 99,124)(100,123)(101,125)
(102,112)(103,111)(104,113)(105,118)(106,117)(107,119)(108,115)(109,114)
(110,116);
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*s2*s3*s2*s3, 
s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s1*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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