Polytope of Type {12,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,4}*768c
if this polytope has a name.
Group : SmallGroup(768,1087653)
Rank : 4
Schlafli Type : {12,4,4}
Number of vertices, edges, etc : 12, 48, 16, 8
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4,4}*384b
4-fold quotients : {3,4,4}*192a, {12,4,2}*192b
8-fold quotients : {6,4,2}*96c
16-fold quotients : {3,4,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)( 55, 56)
( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)
( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)( 76, 95)
( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)(100,147)
(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)(108,159)
(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)(116,179)
(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)(124,191)
(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)(132,163)
(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)(140,175)
(141,169)(142,170)(143,172)(144,171);;
s1 := ( 1,177)( 2,179)( 3,178)( 4,180)( 5,185)( 6,187)( 7,186)( 8,188)
( 9,181)( 10,183)( 11,182)( 12,184)( 13,189)( 14,191)( 15,190)( 16,192)
( 17,161)( 18,163)( 19,162)( 20,164)( 21,169)( 22,171)( 23,170)( 24,172)
( 25,165)( 26,167)( 27,166)( 28,168)( 29,173)( 30,175)( 31,174)( 32,176)
( 33,145)( 34,147)( 35,146)( 36,148)( 37,153)( 38,155)( 39,154)( 40,156)
( 41,149)( 42,151)( 43,150)( 44,152)( 45,157)( 46,159)( 47,158)( 48,160)
( 49,129)( 50,131)( 51,130)( 52,132)( 53,137)( 54,139)( 55,138)( 56,140)
( 57,133)( 58,135)( 59,134)( 60,136)( 61,141)( 62,143)( 63,142)( 64,144)
( 65,113)( 66,115)( 67,114)( 68,116)( 69,121)( 70,123)( 71,122)( 72,124)
( 73,117)( 74,119)( 75,118)( 76,120)( 77,125)( 78,127)( 79,126)( 80,128)
( 81, 97)( 82, 99)( 83, 98)( 84,100)( 85,105)( 86,107)( 87,106)( 88,108)
( 89,101)( 90,103)( 91,102)( 92,104)( 93,109)( 94,111)( 95,110)( 96,112);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112)
(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)
(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)
(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)
(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)
(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192);;
s3 := ( 5, 6)( 7, 8)( 9, 11)( 10, 12)( 13, 16)( 14, 15)( 21, 22)( 23, 24)
( 25, 27)( 26, 28)( 29, 32)( 30, 31)( 37, 38)( 39, 40)( 41, 43)( 42, 44)
( 45, 48)( 46, 47)( 53, 54)( 55, 56)( 57, 59)( 58, 60)( 61, 64)( 62, 63)
( 69, 70)( 71, 72)( 73, 75)( 74, 76)( 77, 80)( 78, 79)( 85, 86)( 87, 88)
( 89, 91)( 90, 92)( 93, 96)( 94, 95)(101,102)(103,104)(105,107)(106,108)
(109,112)(110,111)(117,118)(119,120)(121,123)(122,124)(125,128)(126,127)
(133,134)(135,136)(137,139)(138,140)(141,144)(142,143)(149,150)(151,152)
(153,155)(154,156)(157,160)(158,159)(165,166)(167,168)(169,171)(170,172)
(173,176)(174,175)(181,182)(183,184)(185,187)(186,188)(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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s1*s0*s2*s1*s2*s1*s0*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(192)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 51, 52)
( 55, 56)( 57, 61)( 58, 62)( 59, 64)( 60, 63)( 65, 81)( 66, 82)( 67, 84)
( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)( 73, 93)( 74, 94)( 75, 96)
( 76, 95)( 77, 89)( 78, 90)( 79, 92)( 80, 91)( 97,145)( 98,146)( 99,148)
(100,147)(101,149)(102,150)(103,152)(104,151)(105,157)(106,158)(107,160)
(108,159)(109,153)(110,154)(111,156)(112,155)(113,177)(114,178)(115,180)
(116,179)(117,181)(118,182)(119,184)(120,183)(121,189)(122,190)(123,192)
(124,191)(125,185)(126,186)(127,188)(128,187)(129,161)(130,162)(131,164)
(132,163)(133,165)(134,166)(135,168)(136,167)(137,173)(138,174)(139,176)
(140,175)(141,169)(142,170)(143,172)(144,171);
s1 := Sym(192)!( 1,177)( 2,179)( 3,178)( 4,180)( 5,185)( 6,187)( 7,186)
( 8,188)( 9,181)( 10,183)( 11,182)( 12,184)( 13,189)( 14,191)( 15,190)
( 16,192)( 17,161)( 18,163)( 19,162)( 20,164)( 21,169)( 22,171)( 23,170)
( 24,172)( 25,165)( 26,167)( 27,166)( 28,168)( 29,173)( 30,175)( 31,174)
( 32,176)( 33,145)( 34,147)( 35,146)( 36,148)( 37,153)( 38,155)( 39,154)
( 40,156)( 41,149)( 42,151)( 43,150)( 44,152)( 45,157)( 46,159)( 47,158)
( 48,160)( 49,129)( 50,131)( 51,130)( 52,132)( 53,137)( 54,139)( 55,138)
( 56,140)( 57,133)( 58,135)( 59,134)( 60,136)( 61,141)( 62,143)( 63,142)
( 64,144)( 65,113)( 66,115)( 67,114)( 68,116)( 69,121)( 70,123)( 71,122)
( 72,124)( 73,117)( 74,119)( 75,118)( 76,120)( 77,125)( 78,127)( 79,126)
( 80,128)( 81, 97)( 82, 99)( 83, 98)( 84,100)( 85,105)( 86,107)( 87,106)
( 88,108)( 89,101)( 90,103)( 91,102)( 92,104)( 93,109)( 94,111)( 95,110)
( 96,112);
s2 := Sym(192)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)
(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)
(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)
(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)
(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)
(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)
(188,192);
s3 := Sym(192)!( 5, 6)( 7, 8)( 9, 11)( 10, 12)( 13, 16)( 14, 15)( 21, 22)
( 23, 24)( 25, 27)( 26, 28)( 29, 32)( 30, 31)( 37, 38)( 39, 40)( 41, 43)
( 42, 44)( 45, 48)( 46, 47)( 53, 54)( 55, 56)( 57, 59)( 58, 60)( 61, 64)
( 62, 63)( 69, 70)( 71, 72)( 73, 75)( 74, 76)( 77, 80)( 78, 79)( 85, 86)
( 87, 88)( 89, 91)( 90, 92)( 93, 96)( 94, 95)(101,102)(103,104)(105,107)
(106,108)(109,112)(110,111)(117,118)(119,120)(121,123)(122,124)(125,128)
(126,127)(133,134)(135,136)(137,139)(138,140)(141,144)(142,143)(149,150)
(151,152)(153,155)(154,156)(157,160)(158,159)(165,166)(167,168)(169,171)
(170,172)(173,176)(174,175)(181,182)(183,184)(185,187)(186,188)(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,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s2*s1*s0*s2*s1*s2*s1*s0*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope