Polytope of Type {6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*768a
if this polytope has a name.
Group : SmallGroup(768,1087581)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 96, 192, 64
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4}*384a
   4-fold quotients : {6,4}*192a, {6,4}*192b
   8-fold quotients : {6,4}*96
   16-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   32-fold quotients : {3,4}*24, {6,2}*24
   48-fold quotients : {2,4}*16
   64-fold quotients : {3,2}*12
   96-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope