Polytope of Type {4,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,4}*384c
if this polytope has a name.
Group : SmallGroup(384,20163)
Rank : 4
Schlafli Type : {4,6,4}
Number of vertices, edges, etc : 4, 24, 24, 8
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}*192a, {4,6,4}*192d, {4,6,4}*192e
   4-fold quotients : {4,6,2}*96c, {4,3,4}*96
   8-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,6,4}*768g, {4,12,4}*768i, {4,12,4}*768l, {4,12,4}*768m, {4,12,4}*768n, {4,6,8}*768e, {4,6,8}*768g, {4,6,4}*768l
   3-fold covers : {4,18,4}*1152c, {4,6,12}*1152e, {4,6,12}*1152f
   5-fold covers : {4,6,20}*1920c, {4,30,4}*1920c
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope