Polytope of Type {12,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12,6}*1728e
if this polytope has a 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}*864e, {12,6,6}*864g
   3-fold quotients : {4,12,6}*576b, {12,12,2}*576b
   4-fold quotients : {6,6,6}*432f
   6-fold quotients : {2,12,6}*288b, {6,12,2}*288b, {4,6,6}*288b, {12,6,2}*288c
   8-fold quotients : {6,3,6}*216
   9-fold quotients : {4,12,2}*192a
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {2,12,2}*96, {4,6,2}*96a
   24-fold quotients : {2,3,6}*72, {6,3,2}*72
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,3,2}*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)(  4,124)(  5,126)(  6,125)(  7,121)(  8,123)
(  9,122)( 10,109)( 11,111)( 12,110)( 13,115)( 14,117)( 15,116)( 16,112)
( 17,114)( 18,113)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)( 24,134)
( 25,130)( 26,132)( 27,131)( 28,145)( 29,147)( 30,146)( 31,151)( 32,153)
( 33,152)( 34,148)( 35,150)( 36,149)( 37,136)( 38,138)( 39,137)( 40,142)
( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,154)( 47,156)( 48,155)
( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,172)( 56,174)
( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)( 62,177)( 63,176)( 64,163)
( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)( 70,166)( 71,168)( 72,167)
( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)( 80,186)
( 81,185)( 82,199)( 83,201)( 84,200)( 85,205)( 86,207)( 87,206)( 88,202)
( 89,204)( 90,203)( 91,190)( 92,192)( 93,191)( 94,196)( 95,198)( 96,197)
( 97,193)( 98,195)( 99,194)(100,208)(101,210)(102,209)(103,214)(104,216)
(105,215)(106,211)(107,213)(108,212);;
s2 := (  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)( 13, 20)
( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 32)( 29, 31)( 30, 33)
( 34, 35)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)( 43, 53)
( 44, 52)( 45, 54)( 55, 59)( 56, 58)( 57, 60)( 61, 62)( 64, 77)( 65, 76)
( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 80)( 71, 79)( 72, 81)( 82, 86)
( 83, 85)( 84, 87)( 88, 89)( 91,104)( 92,103)( 93,105)( 94,101)( 95,100)
( 96,102)( 97,107)( 98,106)( 99,108)(109,194)(110,193)(111,195)(112,191)
(113,190)(114,192)(115,197)(116,196)(117,198)(118,212)(119,211)(120,213)
(121,209)(122,208)(123,210)(124,215)(125,214)(126,216)(127,203)(128,202)
(129,204)(130,200)(131,199)(132,201)(133,206)(134,205)(135,207)(136,167)
(137,166)(138,168)(139,164)(140,163)(141,165)(142,170)(143,169)(144,171)
(145,185)(146,184)(147,186)(148,182)(149,181)(150,183)(151,188)(152,187)
(153,189)(154,176)(155,175)(156,177)(157,173)(158,172)(159,174)(160,179)
(161,178)(162,180);;
s3 := (  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)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*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, 
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)(  4,124)(  5,126)(  6,125)(  7,121)
(  8,123)(  9,122)( 10,109)( 11,111)( 12,110)( 13,115)( 14,117)( 15,116)
( 16,112)( 17,114)( 18,113)( 19,127)( 20,129)( 21,128)( 22,133)( 23,135)
( 24,134)( 25,130)( 26,132)( 27,131)( 28,145)( 29,147)( 30,146)( 31,151)
( 32,153)( 33,152)( 34,148)( 35,150)( 36,149)( 37,136)( 38,138)( 39,137)
( 40,142)( 41,144)( 42,143)( 43,139)( 44,141)( 45,140)( 46,154)( 47,156)
( 48,155)( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,172)
( 56,174)( 57,173)( 58,178)( 59,180)( 60,179)( 61,175)( 62,177)( 63,176)
( 64,163)( 65,165)( 66,164)( 67,169)( 68,171)( 69,170)( 70,166)( 71,168)
( 72,167)( 73,181)( 74,183)( 75,182)( 76,187)( 77,189)( 78,188)( 79,184)
( 80,186)( 81,185)( 82,199)( 83,201)( 84,200)( 85,205)( 86,207)( 87,206)
( 88,202)( 89,204)( 90,203)( 91,190)( 92,192)( 93,191)( 94,196)( 95,198)
( 96,197)( 97,193)( 98,195)( 99,194)(100,208)(101,210)(102,209)(103,214)
(104,216)(105,215)(106,211)(107,213)(108,212);
s2 := Sym(216)!(  1,  5)(  2,  4)(  3,  6)(  7,  8)( 10, 23)( 11, 22)( 12, 24)
( 13, 20)( 14, 19)( 15, 21)( 16, 26)( 17, 25)( 18, 27)( 28, 32)( 29, 31)
( 30, 33)( 34, 35)( 37, 50)( 38, 49)( 39, 51)( 40, 47)( 41, 46)( 42, 48)
( 43, 53)( 44, 52)( 45, 54)( 55, 59)( 56, 58)( 57, 60)( 61, 62)( 64, 77)
( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 80)( 71, 79)( 72, 81)
( 82, 86)( 83, 85)( 84, 87)( 88, 89)( 91,104)( 92,103)( 93,105)( 94,101)
( 95,100)( 96,102)( 97,107)( 98,106)( 99,108)(109,194)(110,193)(111,195)
(112,191)(113,190)(114,192)(115,197)(116,196)(117,198)(118,212)(119,211)
(120,213)(121,209)(122,208)(123,210)(124,215)(125,214)(126,216)(127,203)
(128,202)(129,204)(130,200)(131,199)(132,201)(133,206)(134,205)(135,207)
(136,167)(137,166)(138,168)(139,164)(140,163)(141,165)(142,170)(143,169)
(144,171)(145,185)(146,184)(147,186)(148,182)(149,181)(150,183)(151,188)
(152,187)(153,189)(154,176)(155,175)(156,177)(157,173)(158,172)(159,174)
(160,179)(161,178)(162,180);
s3 := 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)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)
(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)
(193,196)(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)
(213,216);
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, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*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, 
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