Polytope of Type {16,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,6}*768b
if this polytope has a name.
Group : SmallGroup(768,1088585)
Rank : 3
Schlafli Type : {16,6}
Number of vertices, edges, etc : 64, 192, 24
Order of s₀s₁s₂ : 48
Order of s₀s₁s₂s₁ : 4
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 : {8,6}*384g
   4-fold quotients : {16,6}*192, {4,6}*192b
   8-fold quotients : {8,6}*96, {4,6}*96
   12-fold quotients : {16,2}*64
   16-fold quotients : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
   24-fold quotients : {8,2}*32
   32-fold quotients : {4,3}*24, {2,6}*24
   48-fold quotients : {4,2}*16
   64-fold quotients : {2,3}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope