Polytope of Type {4,6,26}

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

References : None.
to this polytope