Polytope of Type {6,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,4}*384a
Also Known As : {{6,8|2},{8,4|2}}. if this polytope has another name.
Group : SmallGroup(384,12576)
Rank : 4
Schlafli Type : {6,8,4}
Number of vertices, edges, etc : 6, 24, 16, 4
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,8,4,2} of size 768
Vertex Figure Of :
   {2,6,8,4} of size 768
   {3,6,8,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,4,4}*192, {6,8,2}*192
   3-fold quotients : {2,8,4}*128a
   4-fold quotients : {6,2,4}*96, {6,4,2}*96a
   6-fold quotients : {2,4,4}*64, {2,8,2}*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 : {6,8,4}*768a, {6,8,8}*768b, {6,8,8}*768c, {12,8,4}*768d, {6,16,4}*768a, {6,16,4}*768b
   3-fold covers : {18,8,4}*1152a, {6,8,12}*1152a, {6,24,4}*1152a, {6,24,4}*1152c
   5-fold covers : {30,8,4}*1920a, {6,8,20}*1920a, {6,40,4}*1920a
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 := (  1,122)(  2,121)(  3,123)(  4,125)(  5,124)(  6,126)(  7,128)(  8,127)
(  9,129)( 10,131)( 11,130)( 12,132)( 13,137)( 14,136)( 15,138)( 16,134)
( 17,133)( 18,135)( 19,143)( 20,142)( 21,144)( 22,140)( 23,139)( 24,141)
( 25, 98)( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,102)( 31,104)( 32,103)
( 33,105)( 34,107)( 35,106)( 36,108)( 37,113)( 38,112)( 39,114)( 40,110)
( 41,109)( 42,111)( 43,119)( 44,118)( 45,120)( 46,116)( 47,115)( 48,117)
( 49,170)( 50,169)( 51,171)( 52,173)( 53,172)( 54,174)( 55,176)( 56,175)
( 57,177)( 58,179)( 59,178)( 60,180)( 61,185)( 62,184)( 63,186)( 64,182)
( 65,181)( 66,183)( 67,191)( 68,190)( 69,192)( 70,188)( 71,187)( 72,189)
( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)( 80,151)
( 81,153)( 82,155)( 83,154)( 84,156)( 85,161)( 86,160)( 87,162)( 88,158)
( 89,157)( 90,159)( 91,167)( 92,166)( 93,168)( 94,164)( 95,163)( 96,165);;
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)(145,163)(146,164)(147,165)(148,166)
(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)(156,162)
(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)(176,182)
(177,183)(178,184)(179,185)(180,186);;
s3 := (  1, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)(  8, 56)
(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)( 16, 64)
( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)( 24, 72)
( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)( 32, 80)
( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)( 40, 88)
( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)( 48, 96)
( 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*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
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)!(  1,122)(  2,121)(  3,123)(  4,125)(  5,124)(  6,126)(  7,128)
(  8,127)(  9,129)( 10,131)( 11,130)( 12,132)( 13,137)( 14,136)( 15,138)
( 16,134)( 17,133)( 18,135)( 19,143)( 20,142)( 21,144)( 22,140)( 23,139)
( 24,141)( 25, 98)( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,102)( 31,104)
( 32,103)( 33,105)( 34,107)( 35,106)( 36,108)( 37,113)( 38,112)( 39,114)
( 40,110)( 41,109)( 42,111)( 43,119)( 44,118)( 45,120)( 46,116)( 47,115)
( 48,117)( 49,170)( 50,169)( 51,171)( 52,173)( 53,172)( 54,174)( 55,176)
( 56,175)( 57,177)( 58,179)( 59,178)( 60,180)( 61,185)( 62,184)( 63,186)
( 64,182)( 65,181)( 66,183)( 67,191)( 68,190)( 69,192)( 70,188)( 71,187)
( 72,189)( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,152)
( 80,151)( 81,153)( 82,155)( 83,154)( 84,156)( 85,161)( 86,160)( 87,162)
( 88,158)( 89,157)( 90,159)( 91,167)( 92,166)( 93,168)( 94,164)( 95,163)
( 96,165);
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)(145,163)(146,164)(147,165)
(148,166)(149,167)(150,168)(151,157)(152,158)(153,159)(154,160)(155,161)
(156,162)(169,187)(170,188)(171,189)(172,190)(173,191)(174,192)(175,181)
(176,182)(177,183)(178,184)(179,185)(180,186);
s3 := Sym(192)!(  1, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)
(  8, 56)(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 61)( 14, 62)( 15, 63)
( 16, 64)( 17, 65)( 18, 66)( 19, 67)( 20, 68)( 21, 69)( 22, 70)( 23, 71)
( 24, 72)( 25, 73)( 26, 74)( 27, 75)( 28, 76)( 29, 77)( 30, 78)( 31, 79)
( 32, 80)( 33, 81)( 34, 82)( 35, 83)( 36, 84)( 37, 85)( 38, 86)( 39, 87)
( 40, 88)( 41, 89)( 42, 90)( 43, 91)( 44, 92)( 45, 93)( 46, 94)( 47, 95)
( 48, 96)( 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*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope