Polytope of Type {26,4,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {26,4,6}*1248
Also Known As : {{26,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(1248,1329)
Rank : 4
Schlafli Type : {26,4,6}
Number of vertices, edges, etc : 26, 52, 12, 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 : {26,2,6}*624
   3-fold quotients : {26,4,2}*416
   4-fold quotients : {13,2,6}*312, {26,2,3}*312
   6-fold quotients : {26,2,2}*208
   8-fold quotients : {13,2,3}*156
   12-fold quotients : {13,2,2}*104
   13-fold quotients : {2,4,6}*96a
   26-fold quotients : {2,2,6}*48
   39-fold quotients : {2,4,2}*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 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)( 16, 25)
( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)( 31, 36)
( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)( 46, 47)
( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)( 68, 77)
( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 80, 91)( 81, 90)( 82, 89)( 83, 88)
( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)( 98, 99)
(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(119,130)(120,129)
(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)(135,140)
(136,139)(137,138)(145,156)(146,155)(147,154)(148,153)(149,152)(150,151);;
s1 := (  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)( 16, 26)
( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)( 31, 37)
( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)( 46, 48)
( 53, 54)( 55, 65)( 56, 64)( 57, 63)( 58, 62)( 59, 61)( 66, 67)( 68, 78)
( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 79,119)( 80,118)( 81,130)( 82,129)
( 83,128)( 84,127)( 85,126)( 86,125)( 87,124)( 88,123)( 89,122)( 90,121)
( 91,120)( 92,132)( 93,131)( 94,143)( 95,142)( 96,141)( 97,140)( 98,139)
( 99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,145)(106,144)
(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)(114,149)
(115,148)(116,147)(117,146);;
s2 := (  1, 79)(  2, 80)(  3, 81)(  4, 82)(  5, 83)(  6, 84)(  7, 85)(  8, 86)
(  9, 87)( 10, 88)( 11, 89)( 12, 90)( 13, 91)( 14,105)( 15,106)( 16,107)
( 17,108)( 18,109)( 19,110)( 20,111)( 21,112)( 22,113)( 23,114)( 24,115)
( 25,116)( 26,117)( 27, 92)( 28, 93)( 29, 94)( 30, 95)( 31, 96)( 32, 97)
( 33, 98)( 34, 99)( 35,100)( 36,101)( 37,102)( 38,103)( 39,104)( 40,118)
( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)( 48,126)
( 49,127)( 50,128)( 51,129)( 52,130)( 53,144)( 54,145)( 55,146)( 56,147)
( 57,148)( 58,149)( 59,150)( 60,151)( 61,152)( 62,153)( 63,154)( 64,155)
( 65,156)( 66,131)( 67,132)( 68,133)( 69,134)( 70,135)( 71,136)( 72,137)
( 73,138)( 74,139)( 75,140)( 76,141)( 77,142)( 78,143);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(156)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 15, 26)
( 16, 25)( 17, 24)( 18, 23)( 19, 22)( 20, 21)( 28, 39)( 29, 38)( 30, 37)
( 31, 36)( 32, 35)( 33, 34)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 54, 65)( 55, 64)( 56, 63)( 57, 62)( 58, 61)( 59, 60)( 67, 78)
( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 80, 91)( 81, 90)( 82, 89)
( 83, 88)( 84, 87)( 85, 86)( 93,104)( 94,103)( 95,102)( 96,101)( 97,100)
( 98, 99)(106,117)(107,116)(108,115)(109,114)(110,113)(111,112)(119,130)
(120,129)(121,128)(122,127)(123,126)(124,125)(132,143)(133,142)(134,141)
(135,140)(136,139)(137,138)(145,156)(146,155)(147,154)(148,153)(149,152)
(150,151);
s1 := Sym(156)!(  1,  2)(  3, 13)(  4, 12)(  5, 11)(  6, 10)(  7,  9)( 14, 15)
( 16, 26)( 17, 25)( 18, 24)( 19, 23)( 20, 22)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 41)( 42, 52)( 43, 51)( 44, 50)( 45, 49)
( 46, 48)( 53, 54)( 55, 65)( 56, 64)( 57, 63)( 58, 62)( 59, 61)( 66, 67)
( 68, 78)( 69, 77)( 70, 76)( 71, 75)( 72, 74)( 79,119)( 80,118)( 81,130)
( 82,129)( 83,128)( 84,127)( 85,126)( 86,125)( 87,124)( 88,123)( 89,122)
( 90,121)( 91,120)( 92,132)( 93,131)( 94,143)( 95,142)( 96,141)( 97,140)
( 98,139)( 99,138)(100,137)(101,136)(102,135)(103,134)(104,133)(105,145)
(106,144)(107,156)(108,155)(109,154)(110,153)(111,152)(112,151)(113,150)
(114,149)(115,148)(116,147)(117,146);
s2 := Sym(156)!(  1, 79)(  2, 80)(  3, 81)(  4, 82)(  5, 83)(  6, 84)(  7, 85)
(  8, 86)(  9, 87)( 10, 88)( 11, 89)( 12, 90)( 13, 91)( 14,105)( 15,106)
( 16,107)( 17,108)( 18,109)( 19,110)( 20,111)( 21,112)( 22,113)( 23,114)
( 24,115)( 25,116)( 26,117)( 27, 92)( 28, 93)( 29, 94)( 30, 95)( 31, 96)
( 32, 97)( 33, 98)( 34, 99)( 35,100)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,118)( 41,119)( 42,120)( 43,121)( 44,122)( 45,123)( 46,124)( 47,125)
( 48,126)( 49,127)( 50,128)( 51,129)( 52,130)( 53,144)( 54,145)( 55,146)
( 56,147)( 57,148)( 58,149)( 59,150)( 60,151)( 61,152)( 62,153)( 63,154)
( 64,155)( 65,156)( 66,131)( 67,132)( 68,133)( 69,134)( 70,135)( 71,136)
( 72,137)( 73,138)( 74,139)( 75,140)( 76,141)( 77,142)( 78,143);
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*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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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