Polytope of Type {6,39,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,39,4}*1872
if this polytope has a name.
Group : SmallGroup(1872,1036)
Rank : 4
Schlafli Type : {6,39,4}
Number of vertices, edges, etc : 6, 117, 78, 4
Order of s₀s₁s₂s₃ : 78
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,39,4}*624
   13-fold quotients : {6,3,4}*144
   39-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 53,105)( 54,106)( 55,107)( 56,108)( 57,109)( 58,110)( 59,111)( 60,112)
( 61,113)( 62,114)( 63,115)( 64,116)( 65,117)( 66,118)( 67,119)( 68,120)
( 69,121)( 70,122)( 71,123)( 72,124)( 73,125)( 74,126)( 75,127)( 76,128)
( 77,129)( 78,130)( 79,131)( 80,132)( 81,133)( 82,134)( 83,135)( 84,136)
( 85,137)( 86,138)( 87,139)( 88,140)( 89,141)( 90,142)( 91,143)( 92,144)
( 93,145)( 94,146)( 95,147)( 96,148)( 97,149)( 98,150)( 99,151)(100,152)
(101,153)(102,154)(103,155)(104,156);;
s1 := (  1, 53)(  2, 54)(  3, 56)(  4, 55)(  5,101)(  6,102)(  7,104)(  8,103)
(  9, 97)( 10, 98)( 11,100)( 12, 99)( 13, 93)( 14, 94)( 15, 96)( 16, 95)
( 17, 89)( 18, 90)( 19, 92)( 20, 91)( 21, 85)( 22, 86)( 23, 88)( 24, 87)
( 25, 81)( 26, 82)( 27, 84)( 28, 83)( 29, 77)( 30, 78)( 31, 80)( 32, 79)
( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 69)( 38, 70)( 39, 72)( 40, 71)
( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 61)( 46, 62)( 47, 64)( 48, 63)
( 49, 57)( 50, 58)( 51, 60)( 52, 59)(107,108)(109,153)(110,154)(111,156)
(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)(119,148)
(120,147)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)(127,140)
(128,139)(129,133)(130,134)(131,136)(132,135);;
s2 := (  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9, 49)( 10, 52)( 11, 51)( 12, 50)
( 13, 45)( 14, 48)( 15, 47)( 16, 46)( 17, 41)( 18, 44)( 19, 43)( 20, 42)
( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 33)( 26, 36)( 27, 35)( 28, 34)
( 30, 32)( 53,109)( 54,112)( 55,111)( 56,110)( 57,105)( 58,108)( 59,107)
( 60,106)( 61,153)( 62,156)( 63,155)( 64,154)( 65,149)( 66,152)( 67,151)
( 68,150)( 69,145)( 70,148)( 71,147)( 72,146)( 73,141)( 74,144)( 75,143)
( 76,142)( 77,137)( 78,140)( 79,139)( 80,138)( 81,133)( 82,136)( 83,135)
( 84,134)( 85,129)( 86,132)( 87,131)( 88,130)( 89,125)( 90,128)( 91,127)
( 92,126)( 93,121)( 94,124)( 95,123)( 96,122)( 97,117)( 98,120)( 99,119)
(100,118)(101,113)(102,116)(103,115)(104,114);;
s3 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156);;
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*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s2*s1*s3*s2*s3*s2*s1*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(156)!( 53,105)( 54,106)( 55,107)( 56,108)( 57,109)( 58,110)( 59,111)
( 60,112)( 61,113)( 62,114)( 63,115)( 64,116)( 65,117)( 66,118)( 67,119)
( 68,120)( 69,121)( 70,122)( 71,123)( 72,124)( 73,125)( 74,126)( 75,127)
( 76,128)( 77,129)( 78,130)( 79,131)( 80,132)( 81,133)( 82,134)( 83,135)
( 84,136)( 85,137)( 86,138)( 87,139)( 88,140)( 89,141)( 90,142)( 91,143)
( 92,144)( 93,145)( 94,146)( 95,147)( 96,148)( 97,149)( 98,150)( 99,151)
(100,152)(101,153)(102,154)(103,155)(104,156);
s1 := Sym(156)!(  1, 53)(  2, 54)(  3, 56)(  4, 55)(  5,101)(  6,102)(  7,104)
(  8,103)(  9, 97)( 10, 98)( 11,100)( 12, 99)( 13, 93)( 14, 94)( 15, 96)
( 16, 95)( 17, 89)( 18, 90)( 19, 92)( 20, 91)( 21, 85)( 22, 86)( 23, 88)
( 24, 87)( 25, 81)( 26, 82)( 27, 84)( 28, 83)( 29, 77)( 30, 78)( 31, 80)
( 32, 79)( 33, 73)( 34, 74)( 35, 76)( 36, 75)( 37, 69)( 38, 70)( 39, 72)
( 40, 71)( 41, 65)( 42, 66)( 43, 68)( 44, 67)( 45, 61)( 46, 62)( 47, 64)
( 48, 63)( 49, 57)( 50, 58)( 51, 60)( 52, 59)(107,108)(109,153)(110,154)
(111,156)(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)
(119,148)(120,147)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(129,133)(130,134)(131,136)(132,135);
s2 := Sym(156)!(  1,  5)(  2,  8)(  3,  7)(  4,  6)(  9, 49)( 10, 52)( 11, 51)
( 12, 50)( 13, 45)( 14, 48)( 15, 47)( 16, 46)( 17, 41)( 18, 44)( 19, 43)
( 20, 42)( 21, 37)( 22, 40)( 23, 39)( 24, 38)( 25, 33)( 26, 36)( 27, 35)
( 28, 34)( 30, 32)( 53,109)( 54,112)( 55,111)( 56,110)( 57,105)( 58,108)
( 59,107)( 60,106)( 61,153)( 62,156)( 63,155)( 64,154)( 65,149)( 66,152)
( 67,151)( 68,150)( 69,145)( 70,148)( 71,147)( 72,146)( 73,141)( 74,144)
( 75,143)( 76,142)( 77,137)( 78,140)( 79,139)( 80,138)( 81,133)( 82,136)
( 83,135)( 84,134)( 85,129)( 86,132)( 87,131)( 88,130)( 89,125)( 90,128)
( 91,127)( 92,126)( 93,121)( 94,124)( 95,123)( 96,122)( 97,117)( 98,120)
( 99,119)(100,118)(101,113)(102,116)(103,115)(104,114);
s3 := Sym(156)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156);
poly := sub<Sym(156)|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*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
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