Polytope of Type {6,12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,12}*1728f
if this polytope has a name.
Group : SmallGroup(1728,37586)
Rank : 4
Schlafli Type : {6,12,12}
Number of vertices, edges, etc : 6, 36, 72, 12
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, {6,6,12}*864g
   3-fold quotients : {6,12,4}*576b, {2,12,12}*576c
   4-fold quotients : {6,6,6}*432f
   6-fold quotients : {2,12,6}*288b, {6,12,2}*288b, {6,6,4}*288b, {2,6,12}*288c
   8-fold quotients : {6,3,6}*216
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {2,6,6}*144c, {6,6,2}*144b
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   24-fold quotients : {2,3,6}*72, {6,3,2}*72
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*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 := (  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);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 25)( 17, 27)( 18, 26)( 28, 31)( 29, 33)( 30, 32)
( 35, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 52)
( 44, 54)( 45, 53)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 76)( 65, 78)
( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 79)( 71, 81)( 72, 80)( 82, 85)
( 83, 87)( 84, 86)( 89, 90)( 91,103)( 92,105)( 93,104)( 94,100)( 95,102)
( 96,101)( 97,106)( 98,108)( 99,107)(109,193)(110,195)(111,194)(112,190)
(113,192)(114,191)(115,196)(116,198)(117,197)(118,211)(119,213)(120,212)
(121,208)(122,210)(123,209)(124,214)(125,216)(126,215)(127,202)(128,204)
(129,203)(130,199)(131,201)(132,200)(133,205)(134,207)(135,206)(136,166)
(137,168)(138,167)(139,163)(140,165)(141,164)(142,169)(143,171)(144,170)
(145,184)(146,186)(147,185)(148,181)(149,183)(150,182)(151,187)(152,189)
(153,188)(154,175)(155,177)(156,176)(157,172)(158,174)(159,173)(160,178)
(161,180)(162,179);;
s2 := (  1,119)(  2,118)(  3,120)(  4,125)(  5,124)(  6,126)(  7,122)(  8,121)
(  9,123)( 10,110)( 11,109)( 12,111)( 13,116)( 14,115)( 15,117)( 16,113)
( 17,112)( 18,114)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)
( 25,131)( 26,130)( 27,132)( 28,146)( 29,145)( 30,147)( 31,152)( 32,151)
( 33,153)( 34,149)( 35,148)( 36,150)( 37,137)( 38,136)( 39,138)( 40,143)
( 41,142)( 42,144)( 43,140)( 44,139)( 45,141)( 46,155)( 47,154)( 48,156)
( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,173)( 56,172)
( 57,174)( 58,179)( 59,178)( 60,180)( 61,176)( 62,175)( 63,177)( 64,164)
( 65,163)( 66,165)( 67,170)( 68,169)( 69,171)( 70,167)( 71,166)( 72,168)
( 73,182)( 74,181)( 75,183)( 76,188)( 77,187)( 78,189)( 79,185)( 80,184)
( 81,186)( 82,200)( 83,199)( 84,201)( 85,206)( 86,205)( 87,207)( 88,203)
( 89,202)( 90,204)( 91,191)( 92,190)( 93,192)( 94,197)( 95,196)( 96,198)
( 97,194)( 98,193)( 99,195)(100,209)(101,208)(102,210)(103,215)(104,214)
(105,216)(106,212)(107,211)(108,213);;
s3 := ( 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);;
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, 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, 
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope