Polytope of Type {4,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*384d
if this polytope has a name.
Group : SmallGroup(384,20163)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 8, 24, 24, 4
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,6,4,2} of size 768
Vertex Figure Of :
   {2,4,6,4} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,3,4}*192b, {4,6,4}*192d, {4,6,4}*192f
   4-fold quotients : {2,6,4}*96c, {4,3,4}*96
   8-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,4}*768h, {4,12,4}*768j, {4,12,4}*768k, {4,12,4}*768o, {4,12,4}*768p, {8,6,4}*768e, {8,6,4}*768g, {4,6,4}*768l
   3-fold covers : {4,18,4}*1152d, {12,6,4}*1152f, {12,6,4}*1152g
   5-fold covers : {20,6,4}*1920d, {4,30,4}*1920d
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope