Polytope of Type {6,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,14771)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 18, 108, 72, 4
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 : {6,12,2}*864a, {6,6,4}*864a
   3-fold quotients : {6,12,4}*576b
   4-fold quotients : {6,6,2}*432a
   6-fold quotients : {6,12,2}*288b, {6,6,4}*288b
   8-fold quotients : {6,3,2}*216
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {6,6,2}*144b
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   24-fold quotients : {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 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161)
(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)(176,186)
(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)(199,208)
(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)(207,215);;
s1 := (  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)(  8, 15)
(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)( 31, 45)
( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)( 51, 53)
( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)( 62, 69)
( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)( 85, 99)
( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)(105,107)
(109,145)(110,146)(111,147)(112,153)(113,151)(114,152)(115,149)(116,150)
(117,148)(118,136)(119,137)(120,138)(121,144)(122,142)(123,143)(124,140)
(125,141)(126,139)(127,154)(128,155)(129,156)(130,162)(131,160)(132,161)
(133,158)(134,159)(135,157)(163,199)(164,200)(165,201)(166,207)(167,205)
(168,206)(169,203)(170,204)(171,202)(172,190)(173,191)(174,192)(175,198)
(176,196)(177,197)(178,194)(179,195)(180,193)(181,208)(182,209)(183,210)
(184,216)(185,214)(186,215)(187,212)(188,213)(189,211);;
s2 := (  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)(  8,116)
(  9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)( 16,133)
( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)( 24,120)
( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)( 32,137)
( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)( 40,154)
( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)( 48,150)
( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)( 56,167)
( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)( 64,184)
( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)( 72,189)
( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)( 80,179)
( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)( 88,196)
( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)( 96,210)
( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)(104,200)
(105,201)(106,205)(107,206)(108,207);;
s3 := (109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)
(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)
(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)
(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)
(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)
(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)
(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161)(164,165)(167,168)(170,171)(172,181)(173,183)(174,182)(175,184)
(176,186)(177,185)(178,187)(179,189)(180,188)(191,192)(194,195)(197,198)
(199,208)(200,210)(201,209)(202,211)(203,213)(204,212)(205,214)(206,216)
(207,215);
s1 := Sym(216)!(  1, 10)(  2, 11)(  3, 12)(  4, 18)(  5, 16)(  6, 17)(  7, 14)
(  8, 15)(  9, 13)( 22, 27)( 23, 25)( 24, 26)( 28, 37)( 29, 38)( 30, 39)
( 31, 45)( 32, 43)( 33, 44)( 34, 41)( 35, 42)( 36, 40)( 49, 54)( 50, 52)
( 51, 53)( 55, 64)( 56, 65)( 57, 66)( 58, 72)( 59, 70)( 60, 71)( 61, 68)
( 62, 69)( 63, 67)( 76, 81)( 77, 79)( 78, 80)( 82, 91)( 83, 92)( 84, 93)
( 85, 99)( 86, 97)( 87, 98)( 88, 95)( 89, 96)( 90, 94)(103,108)(104,106)
(105,107)(109,145)(110,146)(111,147)(112,153)(113,151)(114,152)(115,149)
(116,150)(117,148)(118,136)(119,137)(120,138)(121,144)(122,142)(123,143)
(124,140)(125,141)(126,139)(127,154)(128,155)(129,156)(130,162)(131,160)
(132,161)(133,158)(134,159)(135,157)(163,199)(164,200)(165,201)(166,207)
(167,205)(168,206)(169,203)(170,204)(171,202)(172,190)(173,191)(174,192)
(175,198)(176,196)(177,197)(178,194)(179,195)(180,193)(181,208)(182,209)
(183,210)(184,216)(185,214)(186,215)(187,212)(188,213)(189,211);
s2 := Sym(216)!(  1,112)(  2,113)(  3,114)(  4,109)(  5,110)(  6,111)(  7,115)
(  8,116)(  9,117)( 10,130)( 11,131)( 12,132)( 13,127)( 14,128)( 15,129)
( 16,133)( 17,134)( 18,135)( 19,121)( 20,122)( 21,123)( 22,118)( 23,119)
( 24,120)( 25,124)( 26,125)( 27,126)( 28,139)( 29,140)( 30,141)( 31,136)
( 32,137)( 33,138)( 34,142)( 35,143)( 36,144)( 37,157)( 38,158)( 39,159)
( 40,154)( 41,155)( 42,156)( 43,160)( 44,161)( 45,162)( 46,148)( 47,149)
( 48,150)( 49,145)( 50,146)( 51,147)( 52,151)( 53,152)( 54,153)( 55,166)
( 56,167)( 57,168)( 58,163)( 59,164)( 60,165)( 61,169)( 62,170)( 63,171)
( 64,184)( 65,185)( 66,186)( 67,181)( 68,182)( 69,183)( 70,187)( 71,188)
( 72,189)( 73,175)( 74,176)( 75,177)( 76,172)( 77,173)( 78,174)( 79,178)
( 80,179)( 81,180)( 82,193)( 83,194)( 84,195)( 85,190)( 86,191)( 87,192)
( 88,196)( 89,197)( 90,198)( 91,211)( 92,212)( 93,213)( 94,208)( 95,209)
( 96,210)( 97,214)( 98,215)( 99,216)(100,202)(101,203)(102,204)(103,199)
(104,200)(105,201)(106,205)(107,206)(108,207);
s3 := Sym(216)!(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)
(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)
(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)
(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)
(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)
(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)
(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);
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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*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