Polytope of Type {6,8,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8,4}*384b
if this polytope has a name.
Group : SmallGroup(384,12852)
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
   3-fold quotients : {2,8,4}*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 : {6,8,4}*768a, {6,8,8}*768a, {6,8,8}*768d, {12,8,4}*768b
   3-fold covers : {18,8,4}*1152b, {6,8,12}*1152b, {6,24,4}*1152d, {6,24,4}*1152f
   5-fold covers : {30,8,4}*1920b, {6,8,20}*1920b, {6,40,4}*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 := (  1,122)(  2,121)(  3,123)(  4,125)(  5,124)(  6,126)(  7,131)(  8,130)
(  9,132)( 10,128)( 11,127)( 12,129)( 13,137)( 14,136)( 15,138)( 16,134)
( 17,133)( 18,135)( 19,140)( 20,139)( 21,141)( 22,143)( 23,142)( 24,144)
( 25, 98)( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,102)( 31,107)( 32,106)
( 33,108)( 34,104)( 35,103)( 36,105)( 37,113)( 38,112)( 39,114)( 40,110)
( 41,109)( 42,111)( 43,116)( 44,115)( 45,117)( 46,119)( 47,118)( 48,120)
( 49,170)( 50,169)( 51,171)( 52,173)( 53,172)( 54,174)( 55,179)( 56,178)
( 57,180)( 58,176)( 59,175)( 60,177)( 61,185)( 62,184)( 63,186)( 64,182)
( 65,181)( 66,183)( 67,188)( 68,187)( 69,189)( 70,191)( 71,190)( 72,192)
( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,155)( 80,154)
( 81,156)( 82,152)( 83,151)( 84,153)( 85,161)( 86,160)( 87,162)( 88,158)
( 89,157)( 90,159)( 91,164)( 92,163)( 93,165)( 94,167)( 95,166)( 96,168);;
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, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)(  8, 56)
(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 64)( 14, 65)( 15, 66)( 16, 61)
( 17, 62)( 18, 63)( 19, 70)( 20, 71)( 21, 72)( 22, 67)( 23, 68)( 24, 69)
( 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, 88)( 38, 89)( 39, 90)( 40, 85)
( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)( 48, 93)
( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)(104,152)
(105,153)(106,154)(107,155)(108,156)(109,160)(110,161)(111,162)(112,157)
(113,158)(114,159)(115,166)(116,167)(117,168)(118,163)(119,164)(120,165)
(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)(128,176)
(129,177)(130,178)(131,179)(132,180)(133,184)(134,185)(135,186)(136,181)
(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)(144,189);;
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, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s1*s2*s3*s1*s2*s1*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)!(  1,122)(  2,121)(  3,123)(  4,125)(  5,124)(  6,126)(  7,131)
(  8,130)(  9,132)( 10,128)( 11,127)( 12,129)( 13,137)( 14,136)( 15,138)
( 16,134)( 17,133)( 18,135)( 19,140)( 20,139)( 21,141)( 22,143)( 23,142)
( 24,144)( 25, 98)( 26, 97)( 27, 99)( 28,101)( 29,100)( 30,102)( 31,107)
( 32,106)( 33,108)( 34,104)( 35,103)( 36,105)( 37,113)( 38,112)( 39,114)
( 40,110)( 41,109)( 42,111)( 43,116)( 44,115)( 45,117)( 46,119)( 47,118)
( 48,120)( 49,170)( 50,169)( 51,171)( 52,173)( 53,172)( 54,174)( 55,179)
( 56,178)( 57,180)( 58,176)( 59,175)( 60,177)( 61,185)( 62,184)( 63,186)
( 64,182)( 65,181)( 66,183)( 67,188)( 68,187)( 69,189)( 70,191)( 71,190)
( 72,192)( 73,146)( 74,145)( 75,147)( 76,149)( 77,148)( 78,150)( 79,155)
( 80,154)( 81,156)( 82,152)( 83,151)( 84,153)( 85,161)( 86,160)( 87,162)
( 88,158)( 89,157)( 90,159)( 91,164)( 92,163)( 93,165)( 94,167)( 95,166)
( 96,168);
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, 49)(  2, 50)(  3, 51)(  4, 52)(  5, 53)(  6, 54)(  7, 55)
(  8, 56)(  9, 57)( 10, 58)( 11, 59)( 12, 60)( 13, 64)( 14, 65)( 15, 66)
( 16, 61)( 17, 62)( 18, 63)( 19, 70)( 20, 71)( 21, 72)( 22, 67)( 23, 68)
( 24, 69)( 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, 88)( 38, 89)( 39, 90)
( 40, 85)( 41, 86)( 42, 87)( 43, 94)( 44, 95)( 45, 96)( 46, 91)( 47, 92)
( 48, 93)( 97,145)( 98,146)( 99,147)(100,148)(101,149)(102,150)(103,151)
(104,152)(105,153)(106,154)(107,155)(108,156)(109,160)(110,161)(111,162)
(112,157)(113,158)(114,159)(115,166)(116,167)(117,168)(118,163)(119,164)
(120,165)(121,169)(122,170)(123,171)(124,172)(125,173)(126,174)(127,175)
(128,176)(129,177)(130,178)(131,179)(132,180)(133,184)(134,185)(135,186)
(136,181)(137,182)(138,183)(139,190)(140,191)(141,192)(142,187)(143,188)
(144,189);
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, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope