Polytope of Type {32,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {32,4}*256b
if this polytope has a name.
Group : SmallGroup(256,6655)
Rank : 3
Schlafli Type : {32,4}
Number of vertices, edges, etc : 32, 64, 4
Order of s₀s₁s₂ : 32
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {32,4,2} of size 512
Vertex Figure Of :
   {2,32,4} of size 512
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {16,4}*128a
   4-fold quotients : {8,4}*64a, {16,2}*64
   8-fold quotients : {4,4}*32, {8,2}*32
   16-fold quotients : {2,4}*16, {4,2}*16
   32-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {32,4}*512a, {32,8}*512c, {32,8}*512d
   3-fold covers : {32,12}*768b, {96,4}*768b
   5-fold covers : {32,20}*1280b, {160,4}*1280b
   7-fold covers : {32,28}*1792b, {224,4}*1792b
Permutation Representation (GAP) :

s0 := (  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)(  8, 70)
(  9, 74)( 10, 73)( 11, 75)( 12, 76)( 13, 80)( 14, 79)( 15, 78)( 16, 77)
( 17, 85)( 18, 86)( 19, 88)( 20, 87)( 21, 81)( 22, 82)( 23, 84)( 24, 83)
( 25, 94)( 26, 93)( 27, 95)( 28, 96)( 29, 90)( 30, 89)( 31, 91)( 32, 92)
( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,103)( 38,104)( 39,101)( 40,102)
( 41,106)( 42,105)( 43,107)( 44,108)( 45,112)( 46,111)( 47,110)( 48,109)
( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)( 56,115)
( 57,126)( 58,125)( 59,127)( 60,128)( 61,122)( 62,121)( 63,123)( 64,124);;
s1 := (  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 17, 21)( 18, 22)
( 19, 24)( 20, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 61)( 50, 62)
( 51, 64)( 52, 63)( 53, 57)( 54, 58)( 55, 60)( 56, 59)( 65, 81)( 66, 82)
( 67, 84)( 68, 83)( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 73, 89)( 74, 90)
( 75, 92)( 76, 91)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,122)( 98,121)
( 99,123)(100,124)(101,128)(102,127)(103,126)(104,125)(105,114)(106,113)
(107,115)(108,116)(109,120)(110,119)(111,118)(112,117);;
s2 := (  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, 50)( 18, 49)( 19, 52)( 20, 51)( 21, 54)( 22, 53)( 23, 56)( 24, 55)
( 25, 58)( 26, 57)( 27, 60)( 28, 59)( 29, 62)( 30, 61)( 31, 64)( 32, 63)
( 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)
( 81,114)( 82,113)( 83,116)( 84,115)( 85,118)( 86,117)( 87,120)( 88,119)
( 89,122)( 90,121)( 91,124)( 92,123)( 93,126)( 94,125)( 95,128)( 96,127);;
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, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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(128)!(  1, 65)(  2, 66)(  3, 68)(  4, 67)(  5, 71)(  6, 72)(  7, 69)
(  8, 70)(  9, 74)( 10, 73)( 11, 75)( 12, 76)( 13, 80)( 14, 79)( 15, 78)
( 16, 77)( 17, 85)( 18, 86)( 19, 88)( 20, 87)( 21, 81)( 22, 82)( 23, 84)
( 24, 83)( 25, 94)( 26, 93)( 27, 95)( 28, 96)( 29, 90)( 30, 89)( 31, 91)
( 32, 92)( 33, 97)( 34, 98)( 35,100)( 36, 99)( 37,103)( 38,104)( 39,101)
( 40,102)( 41,106)( 42,105)( 43,107)( 44,108)( 45,112)( 46,111)( 47,110)
( 48,109)( 49,117)( 50,118)( 51,120)( 52,119)( 53,113)( 54,114)( 55,116)
( 56,115)( 57,126)( 58,125)( 59,127)( 60,128)( 61,122)( 62,121)( 63,123)
( 64,124);
s1 := Sym(128)!(  3,  4)(  5,  7)(  6,  8)( 11, 12)( 13, 15)( 14, 16)( 17, 21)
( 18, 22)( 19, 24)( 20, 23)( 25, 29)( 26, 30)( 27, 32)( 28, 31)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 49, 61)
( 50, 62)( 51, 64)( 52, 63)( 53, 57)( 54, 58)( 55, 60)( 56, 59)( 65, 81)
( 66, 82)( 67, 84)( 68, 83)( 69, 87)( 70, 88)( 71, 85)( 72, 86)( 73, 89)
( 74, 90)( 75, 92)( 76, 91)( 77, 95)( 78, 96)( 79, 93)( 80, 94)( 97,122)
( 98,121)( 99,123)(100,124)(101,128)(102,127)(103,126)(104,125)(105,114)
(106,113)(107,115)(108,116)(109,120)(110,119)(111,118)(112,117);
s2 := 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, 50)( 18, 49)( 19, 52)( 20, 51)( 21, 54)( 22, 53)( 23, 56)
( 24, 55)( 25, 58)( 26, 57)( 27, 60)( 28, 59)( 29, 62)( 30, 61)( 31, 64)
( 32, 63)( 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)( 81,114)( 82,113)( 83,116)( 84,115)( 85,118)( 86,117)( 87,120)
( 88,119)( 89,122)( 90,121)( 91,124)( 92,123)( 93,126)( 94,125)( 95,128)
( 96,127);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*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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