Polytope of Type {12,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12,6}*1728b
Also Known As : {{12,12|2},{12,6|2}}. if this polytope has another name.
Group : SmallGroup(1728,37586)
Rank : 4
Schlafli Type : {12,12,6}
Number of vertices, edges, etc : 12, 72, 36, 6
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 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 : {6,12,6}*864b, {12,6,6}*864b
   3-fold quotients : {4,12,6}*576a, {12,4,6}*576, {12,12,2}*576a
   4-fold quotients : {6,6,6}*432b
   6-fold quotients : {2,12,6}*288a, {6,12,2}*288a, {12,2,6}*288, {12,6,2}*288a, {4,6,6}*288a, {6,4,6}*288
   9-fold quotients : {4,12,2}*192a, {12,4,2}*192a, {4,4,6}*192
   12-fold quotients : {12,2,3}*144, {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
   18-fold quotients : {2,12,2}*96, {12,2,2}*96, {2,4,6}*96a, {4,2,6}*96, {4,6,2}*96a, {6,4,2}*96a
   24-fold quotients : {3,2,6}*72, {6,2,3}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
   48-fold quotients : {3,2,3}*36
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)(120,156)
(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)(128,146)
(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)
(188,206)(189,207);;
s1 := (  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)(  8,126)
(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)( 16,115)
( 17,117)( 18,116)( 19,127)( 20,129)( 21,128)( 22,130)( 23,132)( 24,131)
( 25,133)( 26,135)( 27,134)( 28,145)( 29,147)( 30,146)( 31,148)( 32,150)
( 33,149)( 34,151)( 35,153)( 36,152)( 37,136)( 38,138)( 39,137)( 40,139)
( 41,141)( 42,140)( 43,142)( 44,144)( 45,143)( 46,154)( 47,156)( 48,155)
( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,172)( 56,174)
( 57,173)( 58,175)( 59,177)( 60,176)( 61,178)( 62,180)( 63,179)( 64,163)
( 65,165)( 66,164)( 67,166)( 68,168)( 69,167)( 70,169)( 71,171)( 72,170)
( 73,181)( 74,183)( 75,182)( 76,184)( 77,186)( 78,185)( 79,187)( 80,189)
( 81,188)( 82,199)( 83,201)( 84,200)( 85,202)( 86,204)( 87,203)( 88,205)
( 89,207)( 90,206)( 91,190)( 92,192)( 93,191)( 94,193)( 95,195)( 96,194)
( 97,196)( 98,198)( 99,197)(100,208)(101,210)(102,209)(103,211)(104,213)
(105,212)(106,214)(107,216)(108,215);;
s2 := (  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 11)( 13, 17)( 14, 16)( 15, 18)
( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 29)( 31, 35)( 32, 34)( 33, 36)
( 37, 38)( 40, 44)( 41, 43)( 42, 45)( 46, 47)( 49, 53)( 50, 52)( 51, 54)
( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 65)( 67, 71)( 68, 70)( 69, 72)
( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 83)( 85, 89)( 86, 88)( 87, 90)
( 91, 92)( 94, 98)( 95, 97)( 96, 99)(100,101)(103,107)(104,106)(105,108)
(109,164)(110,163)(111,165)(112,170)(113,169)(114,171)(115,167)(116,166)
(117,168)(118,173)(119,172)(120,174)(121,179)(122,178)(123,180)(124,176)
(125,175)(126,177)(127,182)(128,181)(129,183)(130,188)(131,187)(132,189)
(133,185)(134,184)(135,186)(136,191)(137,190)(138,192)(139,197)(140,196)
(141,198)(142,194)(143,193)(144,195)(145,200)(146,199)(147,201)(148,206)
(149,205)(150,207)(151,203)(152,202)(153,204)(154,209)(155,208)(156,210)
(157,215)(158,214)(159,216)(160,212)(161,211)(162,213);;
s3 := (  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)(164,167)
(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)(190,193)
(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)(210,213);;
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*s2*s1*s0*s1*s2*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(216)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)
(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)
(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)
(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207);
s1 := Sym(216)!(  1,118)(  2,120)(  3,119)(  4,121)(  5,123)(  6,122)(  7,124)
(  8,126)(  9,125)( 10,109)( 11,111)( 12,110)( 13,112)( 14,114)( 15,113)
( 16,115)( 17,117)( 18,116)( 19,127)( 20,129)( 21,128)( 22,130)( 23,132)
( 24,131)( 25,133)( 26,135)( 27,134)( 28,145)( 29,147)( 30,146)( 31,148)
( 32,150)( 33,149)( 34,151)( 35,153)( 36,152)( 37,136)( 38,138)( 39,137)
( 40,139)( 41,141)( 42,140)( 43,142)( 44,144)( 45,143)( 46,154)( 47,156)
( 48,155)( 49,157)( 50,159)( 51,158)( 52,160)( 53,162)( 54,161)( 55,172)
( 56,174)( 57,173)( 58,175)( 59,177)( 60,176)( 61,178)( 62,180)( 63,179)
( 64,163)( 65,165)( 66,164)( 67,166)( 68,168)( 69,167)( 70,169)( 71,171)
( 72,170)( 73,181)( 74,183)( 75,182)( 76,184)( 77,186)( 78,185)( 79,187)
( 80,189)( 81,188)( 82,199)( 83,201)( 84,200)( 85,202)( 86,204)( 87,203)
( 88,205)( 89,207)( 90,206)( 91,190)( 92,192)( 93,191)( 94,193)( 95,195)
( 96,194)( 97,196)( 98,198)( 99,197)(100,208)(101,210)(102,209)(103,211)
(104,213)(105,212)(106,214)(107,216)(108,215);
s2 := Sym(216)!(  1,  2)(  4,  8)(  5,  7)(  6,  9)( 10, 11)( 13, 17)( 14, 16)
( 15, 18)( 19, 20)( 22, 26)( 23, 25)( 24, 27)( 28, 29)( 31, 35)( 32, 34)
( 33, 36)( 37, 38)( 40, 44)( 41, 43)( 42, 45)( 46, 47)( 49, 53)( 50, 52)
( 51, 54)( 55, 56)( 58, 62)( 59, 61)( 60, 63)( 64, 65)( 67, 71)( 68, 70)
( 69, 72)( 73, 74)( 76, 80)( 77, 79)( 78, 81)( 82, 83)( 85, 89)( 86, 88)
( 87, 90)( 91, 92)( 94, 98)( 95, 97)( 96, 99)(100,101)(103,107)(104,106)
(105,108)(109,164)(110,163)(111,165)(112,170)(113,169)(114,171)(115,167)
(116,166)(117,168)(118,173)(119,172)(120,174)(121,179)(122,178)(123,180)
(124,176)(125,175)(126,177)(127,182)(128,181)(129,183)(130,188)(131,187)
(132,189)(133,185)(134,184)(135,186)(136,191)(137,190)(138,192)(139,197)
(140,196)(141,198)(142,194)(143,193)(144,195)(145,200)(146,199)(147,201)
(148,206)(149,205)(150,207)(151,203)(152,202)(153,204)(154,209)(155,208)
(156,210)(157,215)(158,214)(159,216)(160,212)(161,211)(162,213);
s3 := Sym(216)!(  1,  4)(  2,  5)(  3,  6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,166)
(164,167)(165,168)(172,175)(173,176)(174,177)(181,184)(182,185)(183,186)
(190,193)(191,194)(192,195)(199,202)(200,203)(201,204)(208,211)(209,212)
(210,213);
poly := sub<Sym(216)|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*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

References : None.
to this polytope