Polytope of Type {12,4,6,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4,6,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,37587)
Rank : 5
Schlafli Type : {12,4,6,3}
Number of vertices, edges, etc : 12, 24, 12, 9, 3
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
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 : {12,2,6,3}*864, {6,4,6,3}*864
3-fold quotients : {12,4,2,3}*576a, {4,4,6,3}*576
4-fold quotients : {6,2,6,3}*432
6-fold quotients : {12,2,2,3}*288, {4,2,6,3}*288, {6,4,2,3}*288a, {2,4,6,3}*288
8-fold quotients : {3,2,6,3}*216
9-fold quotients : {4,4,2,3}*192
12-fold quotients : {2,2,6,3}*144, {6,2,2,3}*144
18-fold quotients : {2,4,2,3}*96, {4,2,2,3}*96
24-fold quotients : {3,2,2,3}*72
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)
( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)( 71, 72)
( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)( 95, 96)
( 98, 99)(101,102)(104,105)(107,108)(109,163)(110,165)(111,164)(112,166)
(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)(119,174)(120,173)
(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,181)(128,183)
(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)(136,190)
(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)
(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)(151,205)(152,207)
(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)(160,214)
(161,216)(162,215);;
s1 := ( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)( 8,115)
( 9,117)( 10,119)( 11,118)( 12,120)( 13,122)( 14,121)( 15,123)( 16,125)
( 17,124)( 18,126)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)( 24,132)
( 25,134)( 26,133)( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)( 32,139)
( 33,141)( 34,143)( 35,142)( 36,144)( 37,146)( 38,145)( 39,147)( 40,149)
( 41,148)( 42,150)( 43,152)( 44,151)( 45,153)( 46,155)( 47,154)( 48,156)
( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,164)( 56,163)
( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)( 64,173)
( 65,172)( 66,174)( 67,176)( 68,175)( 69,177)( 70,179)( 71,178)( 72,180)
( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)( 80,187)
( 81,189)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)( 88,197)
( 89,196)( 90,198)( 91,200)( 92,199)( 93,201)( 94,203)( 95,202)( 96,204)
( 97,206)( 98,205)( 99,207)(100,209)(101,208)(102,210)(103,212)(104,211)
(105,213)(106,215)(107,214)(108,216);;
s2 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)
(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)
(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)( 68, 74)
( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)( 91,103)
( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)( 99,108)
(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)(122,128)
(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)(145,157)
(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)
(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)(176,182)
(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)(199,211)
(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)(207,216);;
s4 := ( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)( 8, 14)
( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)( 31, 43)
( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)( 51, 54)
( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)( 62, 68)
( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)( 85, 97)
( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)(105,108)
(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)(116,122)
(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)(139,151)
(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)(159,162)
(163,172)(164,173)(165,174)(166,178)(167,179)(168,180)(169,175)(170,176)
(171,177)(184,187)(185,188)(186,189)(190,199)(191,200)(192,201)(193,205)
(194,206)(195,207)(196,202)(197,203)(198,204)(211,214)(212,215)(213,216);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3,
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(216)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)
( 47, 48)( 50, 51)( 53, 54)( 56, 57)( 59, 60)( 62, 63)( 65, 66)( 68, 69)
( 71, 72)( 74, 75)( 77, 78)( 80, 81)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,163)(110,165)(111,164)
(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)(119,174)
(120,173)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,181)
(128,183)(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)
(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)
(144,197)(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)(151,205)
(152,207)(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)
(160,214)(161,216)(162,215);
s1 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,113)( 5,112)( 6,114)( 7,116)
( 8,115)( 9,117)( 10,119)( 11,118)( 12,120)( 13,122)( 14,121)( 15,123)
( 16,125)( 17,124)( 18,126)( 19,128)( 20,127)( 21,129)( 22,131)( 23,130)
( 24,132)( 25,134)( 26,133)( 27,135)( 28,137)( 29,136)( 30,138)( 31,140)
( 32,139)( 33,141)( 34,143)( 35,142)( 36,144)( 37,146)( 38,145)( 39,147)
( 40,149)( 41,148)( 42,150)( 43,152)( 44,151)( 45,153)( 46,155)( 47,154)
( 48,156)( 49,158)( 50,157)( 51,159)( 52,161)( 53,160)( 54,162)( 55,164)
( 56,163)( 57,165)( 58,167)( 59,166)( 60,168)( 61,170)( 62,169)( 63,171)
( 64,173)( 65,172)( 66,174)( 67,176)( 68,175)( 69,177)( 70,179)( 71,178)
( 72,180)( 73,182)( 74,181)( 75,183)( 76,185)( 77,184)( 78,186)( 79,188)
( 80,187)( 81,189)( 82,191)( 83,190)( 84,192)( 85,194)( 86,193)( 87,195)
( 88,197)( 89,196)( 90,198)( 91,200)( 92,199)( 93,201)( 94,203)( 95,202)
( 96,204)( 97,206)( 98,205)( 99,207)(100,209)(101,208)(102,210)(103,212)
(104,211)(105,213)(106,215)(107,214)(108,216);
s2 := Sym(216)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213);
s3 := Sym(216)!( 1, 4)( 2, 5)( 3, 6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)
( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)
( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)
( 99,108)(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)
(122,128)(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)
(145,157)(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)
(153,162)(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)
(176,182)(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)
(199,211)(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)
(207,216);
s4 := Sym(216)!( 1, 10)( 2, 11)( 3, 12)( 4, 16)( 5, 17)( 6, 18)( 7, 13)
( 8, 14)( 9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)
( 31, 43)( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)
( 51, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)
( 62, 68)( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)
( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)
(105,108)(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)
(116,122)(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)
(139,151)(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)
(159,162)(163,172)(164,173)(165,174)(166,178)(167,179)(168,180)(169,175)
(170,176)(171,177)(184,187)(185,188)(186,189)(190,199)(191,200)(192,201)
(193,205)(194,206)(195,207)(196,202)(197,203)(198,204)(211,214)(212,215)
(213,216);
poly := sub<Sym(216)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 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