Polytope of Type {4,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,12}*384
Also Known As : {{4,4|2},{4,12|2}}. if this polytope has another name.
Group : SmallGroup(384,7460)
Rank : 4
Schlafli Type : {4,4,12}
Number of vertices, edges, etc : 4, 8, 24, 12
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,4,12,2} of size 768
Vertex Figure Of :
   {2,4,4,12} of size 768
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,4,12}*192a, {4,2,12}*192, {4,4,6}*192
   3-fold quotients : {4,4,4}*128
   4-fold quotients : {2,2,12}*96, {2,4,6}*96a, {4,2,6}*96
   6-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
   8-fold quotients : {4,2,3}*48, {2,2,6}*48
   12-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
   16-fold quotients : {2,2,3}*24
   24-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,4,12}*768a, {4,4,24}*768a, {8,4,12}*768b, {4,4,24}*768b, {4,8,12}*768a, {4,4,12}*768a, {4,4,12}*768b, {4,8,12}*768b, {4,8,12}*768c, {4,8,12}*768d
   3-fold covers : {4,4,36}*1152, {4,12,12}*1152b, {4,12,12}*1152c, {12,4,12}*1152
   5-fold covers : {4,4,60}*1920, {4,20,12}*1920, {20,4,12}*1920
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope