Polytope of Type {3,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,4}*768b
Also Known As : {{3,6}₈,{6,4|2}}. if this polytope has another name.
Group : SmallGroup(768,1087795)
Rank : 4
Schlafli Type : {3,6,4}
Number of vertices, edges, etc : 16, 48, 64, 4
Order of s₀s₁s₂s₃ : 8
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   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) :
   2-fold quotients : {3,6,2}*384
   4-fold quotients : {3,6,4}*192
   8-fold quotients : {3,6,2}*96
   16-fold quotients : {3,3,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 53, 54)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)(106,110)
(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)(118,133)
(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)(156,159)
(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)(168,184)
(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187);;
s1 := (  1, 33)(  2, 35)(  3, 34)(  4, 36)(  5, 44)(  6, 42)(  7, 43)(  8, 41)
(  9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)( 16, 48)
( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)( 50, 83)
( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)( 58, 86)
( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)( 69, 76)
( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)(100,132)
(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)(108,133)
(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)(119,123)
(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)(150,186)
(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)(158,191)
(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)(174,175);;
s2 := (  1,103)(  2,104)(  3,102)(  4,101)(  5,100)(  6, 99)(  7, 97)(  8, 98)
(  9,105)( 10,106)( 11,108)( 12,107)( 13,109)( 14,110)( 15,112)( 16,111)
( 17,135)( 18,136)( 19,134)( 20,133)( 21,132)( 22,131)( 23,129)( 24,130)
( 25,137)( 26,138)( 27,140)( 28,139)( 29,141)( 30,142)( 31,144)( 32,143)
( 33,119)( 34,120)( 35,118)( 36,117)( 37,116)( 38,115)( 39,113)( 40,114)
( 41,121)( 42,122)( 43,124)( 44,123)( 45,125)( 46,126)( 47,128)( 48,127)
( 49,151)( 50,152)( 51,150)( 52,149)( 53,148)( 54,147)( 55,145)( 56,146)
( 57,153)( 58,154)( 59,156)( 60,155)( 61,157)( 62,158)( 63,160)( 64,159)
( 65,183)( 66,184)( 67,182)( 68,181)( 69,180)( 70,179)( 71,177)( 72,178)
( 73,185)( 74,186)( 75,188)( 76,187)( 77,189)( 78,190)( 79,192)( 80,191)
( 81,167)( 82,168)( 83,166)( 84,165)( 85,164)( 86,163)( 87,161)( 88,162)
( 89,169)( 90,170)( 91,172)( 92,171)( 93,173)( 94,174)( 95,176)( 96,175);;
s3 := ( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)(112,160)
(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)(136,184)
(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)(144,192);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(192)!(  3,  4)(  5,  6)(  9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 53, 54)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 86)( 70, 85)( 71, 87)( 72, 88)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 99,100)(101,102)(105,109)
(106,110)(107,112)(108,111)(113,129)(114,130)(115,132)(116,131)(117,134)
(118,133)(119,135)(120,136)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(147,148)(149,150)(153,157)(154,158)(155,160)
(156,159)(161,177)(162,178)(163,180)(164,179)(165,182)(166,181)(167,183)
(168,184)(169,189)(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)
(176,187);
s1 := Sym(192)!(  1, 33)(  2, 35)(  3, 34)(  4, 36)(  5, 44)(  6, 42)(  7, 43)
(  8, 41)(  9, 40)( 10, 38)( 11, 39)( 12, 37)( 13, 45)( 14, 47)( 15, 46)
( 16, 48)( 18, 19)( 21, 28)( 22, 26)( 23, 27)( 24, 25)( 30, 31)( 49, 81)
( 50, 83)( 51, 82)( 52, 84)( 53, 92)( 54, 90)( 55, 91)( 56, 89)( 57, 88)
( 58, 86)( 59, 87)( 60, 85)( 61, 93)( 62, 95)( 63, 94)( 64, 96)( 66, 67)
( 69, 76)( 70, 74)( 71, 75)( 72, 73)( 78, 79)( 97,129)( 98,131)( 99,130)
(100,132)(101,140)(102,138)(103,139)(104,137)(105,136)(106,134)(107,135)
(108,133)(109,141)(110,143)(111,142)(112,144)(114,115)(117,124)(118,122)
(119,123)(120,121)(126,127)(145,177)(146,179)(147,178)(148,180)(149,188)
(150,186)(151,187)(152,185)(153,184)(154,182)(155,183)(156,181)(157,189)
(158,191)(159,190)(160,192)(162,163)(165,172)(166,170)(167,171)(168,169)
(174,175);
s2 := Sym(192)!(  1,103)(  2,104)(  3,102)(  4,101)(  5,100)(  6, 99)(  7, 97)
(  8, 98)(  9,105)( 10,106)( 11,108)( 12,107)( 13,109)( 14,110)( 15,112)
( 16,111)( 17,135)( 18,136)( 19,134)( 20,133)( 21,132)( 22,131)( 23,129)
( 24,130)( 25,137)( 26,138)( 27,140)( 28,139)( 29,141)( 30,142)( 31,144)
( 32,143)( 33,119)( 34,120)( 35,118)( 36,117)( 37,116)( 38,115)( 39,113)
( 40,114)( 41,121)( 42,122)( 43,124)( 44,123)( 45,125)( 46,126)( 47,128)
( 48,127)( 49,151)( 50,152)( 51,150)( 52,149)( 53,148)( 54,147)( 55,145)
( 56,146)( 57,153)( 58,154)( 59,156)( 60,155)( 61,157)( 62,158)( 63,160)
( 64,159)( 65,183)( 66,184)( 67,182)( 68,181)( 69,180)( 70,179)( 71,177)
( 72,178)( 73,185)( 74,186)( 75,188)( 76,187)( 77,189)( 78,190)( 79,192)
( 80,191)( 81,167)( 82,168)( 83,166)( 84,165)( 85,164)( 86,163)( 87,161)
( 88,162)( 89,169)( 90,170)( 91,172)( 92,171)( 93,173)( 94,174)( 95,176)
( 96,175);
s3 := Sym(192)!( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,157)(110,158)(111,159)
(112,160)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)
(120,168)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,181)(134,182)(135,183)
(136,184)(137,185)(138,186)(139,187)(140,188)(141,189)(142,190)(143,191)
(144,192);
poly := sub<Sym(192)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope