Polytope of Type {4,26,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,26,6}*1248
Also Known As : {{4,26|2},{26,6|2}}. if this polytope has another name.
Group : SmallGroup(1248,1329)
Rank : 4
Schlafli Type : {4,26,6}
Number of vertices, edges, etc : 4, 52, 78, 6
Order of s₀s₁s₂s₃ : 156
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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) :
   2-fold quotients : {2,26,6}*624
   3-fold quotients : {4,26,2}*416
   6-fold quotients : {2,26,2}*208
   12-fold quotients : {2,13,2}*104
   13-fold quotients : {4,2,6}*96
   26-fold quotients : {4,2,3}*48, {2,2,6}*48
   39-fold quotients : {4,2,2}*32
   52-fold quotients : {2,2,3}*24
   78-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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