Polytope of Type {12,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,78}*1872d
if this polytope has a name.
Group : SmallGroup(1872,1036)
Rank : 3
Schlafli Type : {12,78}
Number of vertices, edges, etc : 12, 468, 78
Order of s₀s₁s₂ : 39
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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 : {4,78}*624c
   6-fold quotients : {4,39}*312
   13-fold quotients : {12,6}*144d
   39-fold quotients : {4,6}*48b
   78-fold quotients : {4,3}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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