Polytope of Type {6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*768
if this polytope has a name.
Group : SmallGroup(768,1086649)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 16, 192, 64
Order of s₀s₁s₂ : 8
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*384, {6,24}*384a, {6,24}*384b
   4-fold quotients : {6,12}*192a, {6,12}*192b, {6,6}*192b
   8-fold quotients : {3,12}*96, {6,6}*96
   16-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {2,8}*32
   32-fold quotients : {3,3}*24
   48-fold quotients : {2,4}*16
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)( 34, 42)
( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)( 53, 55)
( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)( 63, 69)
( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)( 84, 91)
( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97, 98)(101,104)(102,103)(105,114)
(106,113)(107,115)(108,116)(109,120)(110,119)(111,118)(112,117)(121,122)
(125,128)(126,127)(129,138)(130,137)(131,139)(132,140)(133,144)(134,143)
(135,142)(136,141)(145,146)(149,152)(150,151)(153,162)(154,161)(155,163)
(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)(174,175)
(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)(184,189);;
s1 := (  1, 17)(  2, 18)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 24)(  8, 23)
( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)( 29, 44)
( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 89)( 50, 90)
( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 81)( 58, 82)
( 59, 86)( 60, 85)( 61, 84)( 62, 83)( 63, 88)( 64, 87)( 65, 73)( 66, 74)
( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 80)( 72, 79)( 97,162)( 98,161)
( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,154)(106,153)
(107,157)(108,158)(109,155)(110,156)(111,159)(112,160)(113,146)(114,145)
(115,149)(116,150)(117,147)(118,148)(119,151)(120,152)(121,186)(122,185)
(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,178)(130,177)
(131,181)(132,182)(133,179)(134,180)(135,183)(136,184)(137,170)(138,169)
(139,173)(140,174)(141,171)(142,172)(143,175)(144,176);;
s2 := (  1, 99)(  2,100)(  3, 97)(  4, 98)(  5,102)(  6,101)(  7,103)(  8,104)
(  9,115)( 10,116)( 11,113)( 12,114)( 13,118)( 14,117)( 15,119)( 16,120)
( 17,107)( 18,108)( 19,105)( 20,106)( 21,110)( 22,109)( 23,111)( 24,112)
( 25,123)( 26,124)( 27,121)( 28,122)( 29,126)( 30,125)( 31,127)( 32,128)
( 33,139)( 34,140)( 35,137)( 36,138)( 37,142)( 38,141)( 39,143)( 40,144)
( 41,131)( 42,132)( 43,129)( 44,130)( 45,134)( 46,133)( 47,135)( 48,136)
( 49,171)( 50,172)( 51,169)( 52,170)( 53,174)( 54,173)( 55,175)( 56,176)
( 57,187)( 58,188)( 59,185)( 60,186)( 61,190)( 62,189)( 63,191)( 64,192)
( 65,179)( 66,180)( 67,177)( 68,178)( 69,182)( 70,181)( 71,183)( 72,184)
( 73,147)( 74,148)( 75,145)( 76,146)( 77,150)( 78,149)( 79,151)( 80,152)
( 81,163)( 82,164)( 83,161)( 84,162)( 85,166)( 86,165)( 87,167)( 88,168)
( 89,155)( 90,156)( 91,153)( 92,154)( 93,158)( 94,157)( 95,159)( 96,160);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(192)!(  3,  4)(  5,  7)(  6,  8)(  9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 27, 28)( 29, 31)( 30, 32)( 33, 41)
( 34, 42)( 35, 44)( 36, 43)( 37, 47)( 38, 48)( 39, 45)( 40, 46)( 51, 52)
( 53, 55)( 54, 56)( 57, 65)( 58, 66)( 59, 68)( 60, 67)( 61, 71)( 62, 72)
( 63, 69)( 64, 70)( 75, 76)( 77, 79)( 78, 80)( 81, 89)( 82, 90)( 83, 92)
( 84, 91)( 85, 95)( 86, 96)( 87, 93)( 88, 94)( 97, 98)(101,104)(102,103)
(105,114)(106,113)(107,115)(108,116)(109,120)(110,119)(111,118)(112,117)
(121,122)(125,128)(126,127)(129,138)(130,137)(131,139)(132,140)(133,144)
(134,143)(135,142)(136,141)(145,146)(149,152)(150,151)(153,162)(154,161)
(155,163)(156,164)(157,168)(158,167)(159,166)(160,165)(169,170)(173,176)
(174,175)(177,186)(178,185)(179,187)(180,188)(181,192)(182,191)(183,190)
(184,189);
s1 := Sym(192)!(  1, 17)(  2, 18)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 24)
(  8, 23)( 11, 14)( 12, 13)( 15, 16)( 25, 41)( 26, 42)( 27, 46)( 28, 45)
( 29, 44)( 30, 43)( 31, 48)( 32, 47)( 35, 38)( 36, 37)( 39, 40)( 49, 89)
( 50, 90)( 51, 94)( 52, 93)( 53, 92)( 54, 91)( 55, 96)( 56, 95)( 57, 81)
( 58, 82)( 59, 86)( 60, 85)( 61, 84)( 62, 83)( 63, 88)( 64, 87)( 65, 73)
( 66, 74)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 80)( 72, 79)( 97,162)
( 98,161)( 99,165)(100,166)(101,163)(102,164)(103,167)(104,168)(105,154)
(106,153)(107,157)(108,158)(109,155)(110,156)(111,159)(112,160)(113,146)
(114,145)(115,149)(116,150)(117,147)(118,148)(119,151)(120,152)(121,186)
(122,185)(123,189)(124,190)(125,187)(126,188)(127,191)(128,192)(129,178)
(130,177)(131,181)(132,182)(133,179)(134,180)(135,183)(136,184)(137,170)
(138,169)(139,173)(140,174)(141,171)(142,172)(143,175)(144,176);
s2 := Sym(192)!(  1, 99)(  2,100)(  3, 97)(  4, 98)(  5,102)(  6,101)(  7,103)
(  8,104)(  9,115)( 10,116)( 11,113)( 12,114)( 13,118)( 14,117)( 15,119)
( 16,120)( 17,107)( 18,108)( 19,105)( 20,106)( 21,110)( 22,109)( 23,111)
( 24,112)( 25,123)( 26,124)( 27,121)( 28,122)( 29,126)( 30,125)( 31,127)
( 32,128)( 33,139)( 34,140)( 35,137)( 36,138)( 37,142)( 38,141)( 39,143)
( 40,144)( 41,131)( 42,132)( 43,129)( 44,130)( 45,134)( 46,133)( 47,135)
( 48,136)( 49,171)( 50,172)( 51,169)( 52,170)( 53,174)( 54,173)( 55,175)
( 56,176)( 57,187)( 58,188)( 59,185)( 60,186)( 61,190)( 62,189)( 63,191)
( 64,192)( 65,179)( 66,180)( 67,177)( 68,178)( 69,182)( 70,181)( 71,183)
( 72,184)( 73,147)( 74,148)( 75,145)( 76,146)( 77,150)( 78,149)( 79,151)
( 80,152)( 81,163)( 82,164)( 83,161)( 84,162)( 85,166)( 86,165)( 87,167)
( 88,168)( 89,155)( 90,156)( 91,153)( 92,154)( 93,158)( 94,157)( 95,159)
( 96,160);
poly := sub<Sym(192)|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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1 >;

References : None.
to this polytope