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Polytope of Type {4,16}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,16}*256a
if this polytope has a name.
Group : SmallGroup(256,5090)
Rank : 3
Schlafli Type : {4,16}
Number of vertices, edges, etc : 8, 64, 32
Order of s0s1s2 : 16
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,16,2} of size 512
Vertex Figure Of :
   {2,4,16} of size 512
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,8}*128a, {4,16}*128a, {4,16}*128b
   4-fold quotients : {4,8}*64a, {4,8}*64b, {4,4}*64, {2,16}*64
   8-fold quotients : {4,4}*32, {2,8}*32
   16-fold quotients : {2,4}*16, {4,2}*16
   32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,16}*512a, {8,16}*512a, {8,16}*512c, {4,32}*512a, {4,32}*512b
   3-fold covers : {12,16}*768a, {4,48}*768a
   5-fold covers : {20,16}*1280a, {4,80}*1280a
   7-fold covers : {28,16}*1792a, {4,112}*1792a
Permutation Representation (GAP) :
s0 := (  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)(  8, 40)
(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)( 16, 48)
( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)( 24, 54)
( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)( 32, 62)
( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)( 72,104)
( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)( 80,112)
( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)( 88,118)
( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)( 96,126);;
s1 := (  5,  6)(  7,  8)( 13, 14)( 15, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)
( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 41)( 34, 42)( 35, 43)( 36, 44)
( 37, 46)( 38, 45)( 39, 48)( 40, 47)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 81)( 66, 82)( 67, 83)( 68, 84)
( 69, 86)( 70, 85)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 91)( 76, 92)
( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 97,123)( 98,124)( 99,121)(100,122)
(101,128)(102,127)(103,126)(104,125)(105,115)(106,116)(107,113)(108,114)
(109,120)(110,119)(111,118)(112,117);;
s2 := (  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 72)(  8, 71)
(  9, 75)( 10, 76)( 11, 73)( 12, 74)( 13, 80)( 14, 79)( 15, 78)( 16, 77)
( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)( 24, 84)
( 25, 95)( 26, 96)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 89)( 32, 90)
( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,102)( 38,101)( 39,104)( 40,103)
( 41,107)( 42,108)( 43,105)( 44,106)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)( 56,116)
( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)( 64,122);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(128)!(  1, 33)(  2, 34)(  3, 35)(  4, 36)(  5, 37)(  6, 38)(  7, 39)
(  8, 40)(  9, 41)( 10, 42)( 11, 43)( 12, 44)( 13, 45)( 14, 46)( 15, 47)
( 16, 48)( 17, 51)( 18, 52)( 19, 49)( 20, 50)( 21, 55)( 22, 56)( 23, 53)
( 24, 54)( 25, 59)( 26, 60)( 27, 57)( 28, 58)( 29, 63)( 30, 64)( 31, 61)
( 32, 62)( 65, 97)( 66, 98)( 67, 99)( 68,100)( 69,101)( 70,102)( 71,103)
( 72,104)( 73,105)( 74,106)( 75,107)( 76,108)( 77,109)( 78,110)( 79,111)
( 80,112)( 81,115)( 82,116)( 83,113)( 84,114)( 85,119)( 86,120)( 87,117)
( 88,118)( 89,123)( 90,124)( 91,121)( 92,122)( 93,127)( 94,128)( 95,125)
( 96,126);
s1 := Sym(128)!(  5,  6)(  7,  8)( 13, 14)( 15, 16)( 17, 21)( 18, 22)( 19, 23)
( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)( 33, 41)( 34, 42)( 35, 43)
( 36, 44)( 37, 46)( 38, 45)( 39, 48)( 40, 47)( 49, 61)( 50, 62)( 51, 63)
( 52, 64)( 53, 57)( 54, 58)( 55, 59)( 56, 60)( 65, 81)( 66, 82)( 67, 83)
( 68, 84)( 69, 86)( 70, 85)( 71, 88)( 72, 87)( 73, 89)( 74, 90)( 75, 91)
( 76, 92)( 77, 94)( 78, 93)( 79, 96)( 80, 95)( 97,123)( 98,124)( 99,121)
(100,122)(101,128)(102,127)(103,126)(104,125)(105,115)(106,116)(107,113)
(108,114)(109,120)(110,119)(111,118)(112,117);
s2 := Sym(128)!(  1, 65)(  2, 66)(  3, 67)(  4, 68)(  5, 70)(  6, 69)(  7, 72)
(  8, 71)(  9, 75)( 10, 76)( 11, 73)( 12, 74)( 13, 80)( 14, 79)( 15, 78)
( 16, 77)( 17, 85)( 18, 86)( 19, 87)( 20, 88)( 21, 81)( 22, 82)( 23, 83)
( 24, 84)( 25, 95)( 26, 96)( 27, 93)( 28, 94)( 29, 91)( 30, 92)( 31, 89)
( 32, 90)( 33, 97)( 34, 98)( 35, 99)( 36,100)( 37,102)( 38,101)( 39,104)
( 40,103)( 41,107)( 42,108)( 43,105)( 44,106)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,117)( 50,118)( 51,119)( 52,120)( 53,113)( 54,114)( 55,115)
( 56,116)( 57,127)( 58,128)( 59,125)( 60,126)( 61,123)( 62,124)( 63,121)
( 64,122);
poly := sub<Sym(128)|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, s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*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 >; 
 
References : None.
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