Polytope of Type {6,12,12}

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

References : None.
to this polytope