Polytope of Type {6,4,8}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4,8}*384b
if this polytope has a name.
Group : SmallGroup(384,12852)
Rank : 4
Schlafli Type : {6,4,8}
Number of vertices, edges, etc : 6, 12, 16, 8
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,4,8,2} of size 768
Vertex Figure Of :
   {2,6,4,8} of size 768
   {3,6,4,8} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,4}*192
   3-fold quotients : {2,4,8}*128b
   4-fold quotients : {6,2,4}*96, {6,4,2}*96a
   6-fold quotients : {2,4,4}*64
   8-fold quotients : {3,2,4}*48, {6,2,2}*48
   12-fold quotients : {2,2,4}*32, {2,4,2}*32
   16-fold quotients : {3,2,2}*24
   24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,4,8}*768a, {6,8,8}*768c, {6,8,8}*768d, {12,4,8}*768b
   3-fold covers : {18,4,8}*1152b, {6,12,8}*1152e, {6,12,8}*1152f, {6,4,24}*1152b
   5-fold covers : {30,4,8}*1920b, {6,20,8}*1920b, {6,4,40}*1920b
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope