Polytope of Type {14,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,24}*672
Also Known As : {14,24|2}. if this polytope has another name.
Group : SmallGroup(672,392)
Rank : 3
Schlafli Type : {14,24}
Number of vertices, edges, etc : 14, 168, 24
Order of s₀s₁s₂ : 168
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,24,2} of size 1344
Vertex Figure Of :
   {2,14,24} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {14,12}*336
   3-fold quotients : {14,8}*224
   4-fold quotients : {14,6}*168
   6-fold quotients : {14,4}*112
   7-fold quotients : {2,24}*96
   12-fold quotients : {14,2}*56
   14-fold quotients : {2,12}*48
   21-fold quotients : {2,8}*32
   24-fold quotients : {7,2}*28
   28-fold quotients : {2,6}*24
   42-fold quotients : {2,4}*16
   56-fold quotients : {2,3}*12
   84-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,48}*1344, {28,24}*1344a
Permutation Representation (GAP) :

s0 := (  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)( 53, 54)
( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)( 73, 76)
( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)( 93, 98)
( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)(109,110)
(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)(129,132)
(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166);;
s1 := (  1,  2)(  3,  7)(  4,  6)(  8, 16)(  9, 15)( 10, 21)( 11, 20)( 12, 19)
( 13, 18)( 14, 17)( 22, 23)( 24, 28)( 25, 27)( 29, 37)( 30, 36)( 31, 42)
( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 65)( 44, 64)( 45, 70)( 46, 69)
( 47, 68)( 48, 67)( 49, 66)( 50, 79)( 51, 78)( 52, 84)( 53, 83)( 54, 82)
( 55, 81)( 56, 80)( 57, 72)( 58, 71)( 59, 77)( 60, 76)( 61, 75)( 62, 74)
( 63, 73)( 85,128)( 86,127)( 87,133)( 88,132)( 89,131)( 90,130)( 91,129)
( 92,142)( 93,141)( 94,147)( 95,146)( 96,145)( 97,144)( 98,143)( 99,135)
(100,134)(101,140)(102,139)(103,138)(104,137)(105,136)(106,149)(107,148)
(108,154)(109,153)(110,152)(111,151)(112,150)(113,163)(114,162)(115,168)
(116,167)(117,166)(118,165)(119,164)(120,156)(121,155)(122,161)(123,160)
(124,159)(125,158)(126,157);;
s2 := (  1, 92)(  2, 93)(  3, 94)(  4, 95)(  5, 96)(  6, 97)(  7, 98)(  8, 85)
(  9, 86)( 10, 87)( 11, 88)( 12, 89)( 13, 90)( 14, 91)( 15, 99)( 16,100)
( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,113)( 23,114)( 24,115)
( 25,116)( 26,117)( 27,118)( 28,119)( 29,106)( 30,107)( 31,108)( 32,109)
( 33,110)( 34,111)( 35,112)( 36,120)( 37,121)( 38,122)( 39,123)( 40,124)
( 41,125)( 42,126)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)( 48,160)
( 49,161)( 50,148)( 51,149)( 52,150)( 53,151)( 54,152)( 55,153)( 56,154)
( 57,162)( 58,163)( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)( 64,134)
( 65,135)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,127)( 72,128)
( 73,129)( 74,130)( 75,131)( 76,132)( 77,133)( 78,141)( 79,142)( 80,143)
( 81,144)( 82,145)( 83,146)( 84,147);;
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, s0*s1*s2*s1*s0*s1*s2*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, 
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(168)!(  2,  7)(  3,  6)(  4,  5)(  9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 51, 56)( 52, 55)
( 53, 54)( 58, 63)( 59, 62)( 60, 61)( 65, 70)( 66, 69)( 67, 68)( 72, 77)
( 73, 76)( 74, 75)( 79, 84)( 80, 83)( 81, 82)( 86, 91)( 87, 90)( 88, 89)
( 93, 98)( 94, 97)( 95, 96)(100,105)(101,104)(102,103)(107,112)(108,111)
(109,110)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(128,133)
(129,132)(130,131)(135,140)(136,139)(137,138)(142,147)(143,146)(144,145)
(149,154)(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)
(165,166);
s1 := Sym(168)!(  1,  2)(  3,  7)(  4,  6)(  8, 16)(  9, 15)( 10, 21)( 11, 20)
( 12, 19)( 13, 18)( 14, 17)( 22, 23)( 24, 28)( 25, 27)( 29, 37)( 30, 36)
( 31, 42)( 32, 41)( 33, 40)( 34, 39)( 35, 38)( 43, 65)( 44, 64)( 45, 70)
( 46, 69)( 47, 68)( 48, 67)( 49, 66)( 50, 79)( 51, 78)( 52, 84)( 53, 83)
( 54, 82)( 55, 81)( 56, 80)( 57, 72)( 58, 71)( 59, 77)( 60, 76)( 61, 75)
( 62, 74)( 63, 73)( 85,128)( 86,127)( 87,133)( 88,132)( 89,131)( 90,130)
( 91,129)( 92,142)( 93,141)( 94,147)( 95,146)( 96,145)( 97,144)( 98,143)
( 99,135)(100,134)(101,140)(102,139)(103,138)(104,137)(105,136)(106,149)
(107,148)(108,154)(109,153)(110,152)(111,151)(112,150)(113,163)(114,162)
(115,168)(116,167)(117,166)(118,165)(119,164)(120,156)(121,155)(122,161)
(123,160)(124,159)(125,158)(126,157);
s2 := Sym(168)!(  1, 92)(  2, 93)(  3, 94)(  4, 95)(  5, 96)(  6, 97)(  7, 98)
(  8, 85)(  9, 86)( 10, 87)( 11, 88)( 12, 89)( 13, 90)( 14, 91)( 15, 99)
( 16,100)( 17,101)( 18,102)( 19,103)( 20,104)( 21,105)( 22,113)( 23,114)
( 24,115)( 25,116)( 26,117)( 27,118)( 28,119)( 29,106)( 30,107)( 31,108)
( 32,109)( 33,110)( 34,111)( 35,112)( 36,120)( 37,121)( 38,122)( 39,123)
( 40,124)( 41,125)( 42,126)( 43,155)( 44,156)( 45,157)( 46,158)( 47,159)
( 48,160)( 49,161)( 50,148)( 51,149)( 52,150)( 53,151)( 54,152)( 55,153)
( 56,154)( 57,162)( 58,163)( 59,164)( 60,165)( 61,166)( 62,167)( 63,168)
( 64,134)( 65,135)( 66,136)( 67,137)( 68,138)( 69,139)( 70,140)( 71,127)
( 72,128)( 73,129)( 74,130)( 75,131)( 76,132)( 77,133)( 78,141)( 79,142)
( 80,143)( 81,144)( 82,145)( 83,146)( 84,147);
poly := sub<Sym(168)|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, s0*s1*s2*s1*s0*s1*s2*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, 
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