Polytope of Type {4,12,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,4}*384g
if this polytope has a name.
Group : SmallGroup(384,20080)
Rank : 4
Schlafli Type : {4,12,4}
Number of vertices, edges, etc : 4, 24, 24, 4
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,12,4,2} of size 768
Vertex Figure Of :
{2,4,12,4} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,4}*192d
4-fold quotients : {4,3,4}*96
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,12,4}*768m, {4,12,4}*768p
3-fold covers : {4,36,4}*1152g
5-fold covers : {4,60,4}*1920g
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 := ( 1, 17)( 2, 20)( 3, 19)( 4, 18)( 5, 25)( 6, 28)( 7, 27)( 8, 26)
( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 13, 29)( 14, 32)( 15, 31)( 16, 30)
( 34, 36)( 37, 41)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 65)( 50, 68)
( 51, 67)( 52, 66)( 53, 73)( 54, 76)( 55, 75)( 56, 74)( 57, 69)( 58, 72)
( 59, 71)( 60, 70)( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)( 85, 89)
( 86, 92)( 87, 91)( 88, 90)( 94, 96)( 97,161)( 98,164)( 99,163)(100,162)
(101,169)(102,172)(103,171)(104,170)(105,165)(106,168)(107,167)(108,166)
(109,173)(110,176)(111,175)(112,174)(113,145)(114,148)(115,147)(116,146)
(117,153)(118,156)(119,155)(120,154)(121,149)(122,152)(123,151)(124,150)
(125,157)(126,160)(127,159)(128,158)(129,177)(130,180)(131,179)(132,178)
(133,185)(134,188)(135,187)(136,186)(137,181)(138,184)(139,183)(140,182)
(141,189)(142,192)(143,191)(144,190);;
s2 := ( 1, 97)( 2, 98)( 3,100)( 4, 99)( 5,109)( 6,110)( 7,112)( 8,111)
( 9,105)( 10,106)( 11,108)( 12,107)( 13,101)( 14,102)( 15,104)( 16,103)
( 17,129)( 18,130)( 19,132)( 20,131)( 21,141)( 22,142)( 23,144)( 24,143)
( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)( 32,135)
( 33,113)( 34,114)( 35,116)( 36,115)( 37,125)( 38,126)( 39,128)( 40,127)
( 41,121)( 42,122)( 43,124)( 44,123)( 45,117)( 46,118)( 47,120)( 48,119)
( 49,145)( 50,146)( 51,148)( 52,147)( 53,157)( 54,158)( 55,160)( 56,159)
( 57,153)( 58,154)( 59,156)( 60,155)( 61,149)( 62,150)( 63,152)( 64,151)
( 65,177)( 66,178)( 67,180)( 68,179)( 69,189)( 70,190)( 71,192)( 72,191)
( 73,185)( 74,186)( 75,188)( 76,187)( 77,181)( 78,182)( 79,184)( 80,183)
( 81,161)( 82,162)( 83,164)( 84,163)( 85,173)( 86,174)( 87,176)( 88,175)
( 89,169)( 90,170)( 91,172)( 92,171)( 93,165)( 94,166)( 95,168)( 96,167);;
s3 := ( 1, 51)( 2, 52)( 3, 49)( 4, 50)( 5, 55)( 6, 56)( 7, 53)( 8, 54)
( 9, 59)( 10, 60)( 11, 57)( 12, 58)( 13, 63)( 14, 64)( 15, 61)( 16, 62)
( 17, 67)( 18, 68)( 19, 65)( 20, 66)( 21, 71)( 22, 72)( 23, 69)( 24, 70)
( 25, 75)( 26, 76)( 27, 73)( 28, 74)( 29, 79)( 30, 80)( 31, 77)( 32, 78)
( 33, 83)( 34, 84)( 35, 81)( 36, 82)( 37, 87)( 38, 88)( 39, 85)( 40, 86)
( 41, 91)( 42, 92)( 43, 89)( 44, 90)( 45, 95)( 46, 96)( 47, 93)( 48, 94)
( 97,147)( 98,148)( 99,145)(100,146)(101,151)(102,152)(103,149)(104,150)
(105,155)(106,156)(107,153)(108,154)(109,159)(110,160)(111,157)(112,158)
(113,163)(114,164)(115,161)(116,162)(117,167)(118,168)(119,165)(120,166)
(121,171)(122,172)(123,169)(124,170)(125,175)(126,176)(127,173)(128,174)
(129,179)(130,180)(131,177)(132,178)(133,183)(134,184)(135,181)(136,182)
(137,187)(138,188)(139,185)(140,186)(141,191)(142,192)(143,189)(144,190);;
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1 ];;
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)!( 1, 17)( 2, 20)( 3, 19)( 4, 18)( 5, 25)( 6, 28)( 7, 27)
( 8, 26)( 9, 21)( 10, 24)( 11, 23)( 12, 22)( 13, 29)( 14, 32)( 15, 31)
( 16, 30)( 34, 36)( 37, 41)( 38, 44)( 39, 43)( 40, 42)( 46, 48)( 49, 65)
( 50, 68)( 51, 67)( 52, 66)( 53, 73)( 54, 76)( 55, 75)( 56, 74)( 57, 69)
( 58, 72)( 59, 71)( 60, 70)( 61, 77)( 62, 80)( 63, 79)( 64, 78)( 82, 84)
( 85, 89)( 86, 92)( 87, 91)( 88, 90)( 94, 96)( 97,161)( 98,164)( 99,163)
(100,162)(101,169)(102,172)(103,171)(104,170)(105,165)(106,168)(107,167)
(108,166)(109,173)(110,176)(111,175)(112,174)(113,145)(114,148)(115,147)
(116,146)(117,153)(118,156)(119,155)(120,154)(121,149)(122,152)(123,151)
(124,150)(125,157)(126,160)(127,159)(128,158)(129,177)(130,180)(131,179)
(132,178)(133,185)(134,188)(135,187)(136,186)(137,181)(138,184)(139,183)
(140,182)(141,189)(142,192)(143,191)(144,190);
s2 := Sym(192)!( 1, 97)( 2, 98)( 3,100)( 4, 99)( 5,109)( 6,110)( 7,112)
( 8,111)( 9,105)( 10,106)( 11,108)( 12,107)( 13,101)( 14,102)( 15,104)
( 16,103)( 17,129)( 18,130)( 19,132)( 20,131)( 21,141)( 22,142)( 23,144)
( 24,143)( 25,137)( 26,138)( 27,140)( 28,139)( 29,133)( 30,134)( 31,136)
( 32,135)( 33,113)( 34,114)( 35,116)( 36,115)( 37,125)( 38,126)( 39,128)
( 40,127)( 41,121)( 42,122)( 43,124)( 44,123)( 45,117)( 46,118)( 47,120)
( 48,119)( 49,145)( 50,146)( 51,148)( 52,147)( 53,157)( 54,158)( 55,160)
( 56,159)( 57,153)( 58,154)( 59,156)( 60,155)( 61,149)( 62,150)( 63,152)
( 64,151)( 65,177)( 66,178)( 67,180)( 68,179)( 69,189)( 70,190)( 71,192)
( 72,191)( 73,185)( 74,186)( 75,188)( 76,187)( 77,181)( 78,182)( 79,184)
( 80,183)( 81,161)( 82,162)( 83,164)( 84,163)( 85,173)( 86,174)( 87,176)
( 88,175)( 89,169)( 90,170)( 91,172)( 92,171)( 93,165)( 94,166)( 95,168)
( 96,167);
s3 := Sym(192)!( 1, 51)( 2, 52)( 3, 49)( 4, 50)( 5, 55)( 6, 56)( 7, 53)
( 8, 54)( 9, 59)( 10, 60)( 11, 57)( 12, 58)( 13, 63)( 14, 64)( 15, 61)
( 16, 62)( 17, 67)( 18, 68)( 19, 65)( 20, 66)( 21, 71)( 22, 72)( 23, 69)
( 24, 70)( 25, 75)( 26, 76)( 27, 73)( 28, 74)( 29, 79)( 30, 80)( 31, 77)
( 32, 78)( 33, 83)( 34, 84)( 35, 81)( 36, 82)( 37, 87)( 38, 88)( 39, 85)
( 40, 86)( 41, 91)( 42, 92)( 43, 89)( 44, 90)( 45, 95)( 46, 96)( 47, 93)
( 48, 94)( 97,147)( 98,148)( 99,145)(100,146)(101,151)(102,152)(103,149)
(104,150)(105,155)(106,156)(107,153)(108,154)(109,159)(110,160)(111,157)
(112,158)(113,163)(114,164)(115,161)(116,162)(117,167)(118,168)(119,165)
(120,166)(121,171)(122,172)(123,169)(124,170)(125,175)(126,176)(127,173)
(128,174)(129,179)(130,180)(131,177)(132,178)(133,183)(134,184)(135,181)
(136,182)(137,187)(138,188)(139,185)(140,186)(141,191)(142,192)(143,189)
(144,190);
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*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2,
s2*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s1 >;
References : None.
to this polytope