Polytope of Type {8,3,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,3,4}*384
if this polytope has a name.
Group : SmallGroup(384,20086)
Rank : 4
Schlafli Type : {8,3,4}
Number of vertices, edges, etc : 16, 24, 12, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,3,4,2} of size 768
Vertex Figure Of :
   {2,8,3,4} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,4}*192b
   4-fold quotients : {4,3,4}*96
   8-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,3,4}*768b, {8,6,4}*768e, {8,6,4}*768f
   3-fold covers : {8,9,4}*1152, {24,3,4}*1152
   5-fold covers : {8,15,4}*1920
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope