Polytope of Type {6,6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,4}*384
if this polytope has a name.
Group : SmallGroup(384,20051)
Rank : 4
Schlafli Type : {6,6,4}
Number of vertices, edges, etc : 8, 24, 16, 4
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,6,4,2} of size 768
Vertex Figure Of :
   {2,6,6,4} of size 768
   {3,6,6,4} of size 1920
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,4}*192, {6,6,2}*192
   4-fold quotients : {3,6,2}*96, {6,3,2}*96
   8-fold quotients : {3,3,2}*48
   12-fold quotients : {2,2,4}*32
   24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,12,4}*768a, {12,6,4}*768a, {6,12,4}*768b, {6,6,8}*768, {6,6,4}*768e, {12,6,4}*768b
   3-fold covers : {6,6,12}*1152a, {6,6,4}*1152e, {6,6,4}*1152f
   5-fold covers : {6,6,20}*1920, {6,30,4}*1920, {30,6,4}*1920
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope