Polytope of Type {2,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,4}*1728d
if this polytope has a name.
Group : SmallGroup(1728,46611)
Rank : 4
Schlafli Type : {2,12,4}
Number of vertices, edges, etc : 2, 108, 216, 36
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   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 : {2,6,4}*864b
   3-fold quotients : {2,12,4}*576
   6-fold quotients : {2,6,4}*288
   9-fold quotients : {2,12,4}*192a
   12-fold quotients : {2,6,4}*144
   18-fold quotients : {2,12,2}*96, {2,6,4}*96a
   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 := (1,2);;
s1 := (  3, 57)(  4, 59)(  5, 58)(  6, 63)(  7, 65)(  8, 64)(  9, 60)( 10, 62)
( 11, 61)( 12, 75)( 13, 77)( 14, 76)( 15, 81)( 16, 83)( 17, 82)( 18, 78)
( 19, 80)( 20, 79)( 21, 66)( 22, 68)( 23, 67)( 24, 72)( 25, 74)( 26, 73)
( 27, 69)( 28, 71)( 29, 70)( 30, 84)( 31, 86)( 32, 85)( 33, 90)( 34, 92)
( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,102)( 40,104)( 41,103)( 42,108)
( 43,110)( 44,109)( 45,105)( 46,107)( 47,106)( 48, 93)( 49, 95)( 50, 94)
( 51, 99)( 52,101)( 53,100)( 54, 96)( 55, 98)( 56, 97)(111,192)(112,194)
(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)(120,210)
(121,212)(122,211)(123,216)(124,218)(125,217)(126,213)(127,215)(128,214)
(129,201)(130,203)(131,202)(132,207)(133,209)(134,208)(135,204)(136,206)
(137,205)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)(144,168)
(145,170)(146,169)(147,183)(148,185)(149,184)(150,189)(151,191)(152,190)
(153,186)(154,188)(155,187)(156,174)(157,176)(158,175)(159,180)(160,182)
(161,181)(162,177)(163,179)(164,178);;
s2 := (  3,169)(  4,168)(  5,170)(  6,166)(  7,165)(  8,167)(  9,172)( 10,171)
( 11,173)( 12,181)( 13,180)( 14,182)( 15,178)( 16,177)( 17,179)( 18,175)
( 19,174)( 20,176)( 21,184)( 22,183)( 23,185)( 24,190)( 25,189)( 26,191)
( 27,187)( 28,186)( 29,188)( 30,196)( 31,195)( 32,197)( 33,193)( 34,192)
( 35,194)( 36,199)( 37,198)( 38,200)( 39,208)( 40,207)( 41,209)( 42,205)
( 43,204)( 44,206)( 45,202)( 46,201)( 47,203)( 48,211)( 49,210)( 50,212)
( 51,217)( 52,216)( 53,218)( 54,214)( 55,213)( 56,215)( 57,115)( 58,114)
( 59,116)( 60,112)( 61,111)( 62,113)( 63,118)( 64,117)( 65,119)( 66,127)
( 67,126)( 68,128)( 69,124)( 70,123)( 71,125)( 72,121)( 73,120)( 74,122)
( 75,130)( 76,129)( 77,131)( 78,136)( 79,135)( 80,137)( 81,133)( 82,132)
( 83,134)( 84,142)( 85,141)( 86,143)( 87,139)( 88,138)( 89,140)( 90,145)
( 91,144)( 92,146)( 93,154)( 94,153)( 95,155)( 96,151)( 97,150)( 98,152)
( 99,148)(100,147)(101,149)(102,157)(103,156)(104,158)(105,163)(106,162)
(107,164)(108,160)(109,159)(110,161);;
s3 := (  6, 24)(  7, 25)(  8, 26)(  9, 18)( 10, 19)( 11, 20)( 12, 21)( 13, 22)
( 14, 23)( 33, 51)( 34, 52)( 35, 53)( 36, 45)( 37, 46)( 38, 47)( 39, 48)
( 40, 49)( 41, 50)( 60, 78)( 61, 79)( 62, 80)( 63, 72)( 64, 73)( 65, 74)
( 66, 75)( 67, 76)( 68, 77)( 87,105)( 88,106)( 89,107)( 90, 99)( 91,100)
( 92,101)( 93,102)( 94,103)( 95,104)(111,165)(112,166)(113,167)(114,186)
(115,187)(116,188)(117,180)(118,181)(119,182)(120,183)(121,184)(122,185)
(123,177)(124,178)(125,179)(126,171)(127,172)(128,173)(129,174)(130,175)
(131,176)(132,168)(133,169)(134,170)(135,189)(136,190)(137,191)(138,192)
(139,193)(140,194)(141,213)(142,214)(143,215)(144,207)(145,208)(146,209)
(147,210)(148,211)(149,212)(150,204)(151,205)(152,206)(153,198)(154,199)
(155,200)(156,201)(157,202)(158,203)(159,195)(160,196)(161,197)(162,216)
(163,217)(164,218);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3, s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*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*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(218)!(1,2);
s1 := Sym(218)!(  3, 57)(  4, 59)(  5, 58)(  6, 63)(  7, 65)(  8, 64)(  9, 60)
( 10, 62)( 11, 61)( 12, 75)( 13, 77)( 14, 76)( 15, 81)( 16, 83)( 17, 82)
( 18, 78)( 19, 80)( 20, 79)( 21, 66)( 22, 68)( 23, 67)( 24, 72)( 25, 74)
( 26, 73)( 27, 69)( 28, 71)( 29, 70)( 30, 84)( 31, 86)( 32, 85)( 33, 90)
( 34, 92)( 35, 91)( 36, 87)( 37, 89)( 38, 88)( 39,102)( 40,104)( 41,103)
( 42,108)( 43,110)( 44,109)( 45,105)( 46,107)( 47,106)( 48, 93)( 49, 95)
( 50, 94)( 51, 99)( 52,101)( 53,100)( 54, 96)( 55, 98)( 56, 97)(111,192)
(112,194)(113,193)(114,198)(115,200)(116,199)(117,195)(118,197)(119,196)
(120,210)(121,212)(122,211)(123,216)(124,218)(125,217)(126,213)(127,215)
(128,214)(129,201)(130,203)(131,202)(132,207)(133,209)(134,208)(135,204)
(136,206)(137,205)(138,165)(139,167)(140,166)(141,171)(142,173)(143,172)
(144,168)(145,170)(146,169)(147,183)(148,185)(149,184)(150,189)(151,191)
(152,190)(153,186)(154,188)(155,187)(156,174)(157,176)(158,175)(159,180)
(160,182)(161,181)(162,177)(163,179)(164,178);
s2 := Sym(218)!(  3,169)(  4,168)(  5,170)(  6,166)(  7,165)(  8,167)(  9,172)
( 10,171)( 11,173)( 12,181)( 13,180)( 14,182)( 15,178)( 16,177)( 17,179)
( 18,175)( 19,174)( 20,176)( 21,184)( 22,183)( 23,185)( 24,190)( 25,189)
( 26,191)( 27,187)( 28,186)( 29,188)( 30,196)( 31,195)( 32,197)( 33,193)
( 34,192)( 35,194)( 36,199)( 37,198)( 38,200)( 39,208)( 40,207)( 41,209)
( 42,205)( 43,204)( 44,206)( 45,202)( 46,201)( 47,203)( 48,211)( 49,210)
( 50,212)( 51,217)( 52,216)( 53,218)( 54,214)( 55,213)( 56,215)( 57,115)
( 58,114)( 59,116)( 60,112)( 61,111)( 62,113)( 63,118)( 64,117)( 65,119)
( 66,127)( 67,126)( 68,128)( 69,124)( 70,123)( 71,125)( 72,121)( 73,120)
( 74,122)( 75,130)( 76,129)( 77,131)( 78,136)( 79,135)( 80,137)( 81,133)
( 82,132)( 83,134)( 84,142)( 85,141)( 86,143)( 87,139)( 88,138)( 89,140)
( 90,145)( 91,144)( 92,146)( 93,154)( 94,153)( 95,155)( 96,151)( 97,150)
( 98,152)( 99,148)(100,147)(101,149)(102,157)(103,156)(104,158)(105,163)
(106,162)(107,164)(108,160)(109,159)(110,161);
s3 := Sym(218)!(  6, 24)(  7, 25)(  8, 26)(  9, 18)( 10, 19)( 11, 20)( 12, 21)
( 13, 22)( 14, 23)( 33, 51)( 34, 52)( 35, 53)( 36, 45)( 37, 46)( 38, 47)
( 39, 48)( 40, 49)( 41, 50)( 60, 78)( 61, 79)( 62, 80)( 63, 72)( 64, 73)
( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 87,105)( 88,106)( 89,107)( 90, 99)
( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)(111,165)(112,166)(113,167)
(114,186)(115,187)(116,188)(117,180)(118,181)(119,182)(120,183)(121,184)
(122,185)(123,177)(124,178)(125,179)(126,171)(127,172)(128,173)(129,174)
(130,175)(131,176)(132,168)(133,169)(134,170)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,213)(142,214)(143,215)(144,207)(145,208)
(146,209)(147,210)(148,211)(149,212)(150,204)(151,205)(152,206)(153,198)
(154,199)(155,200)(156,201)(157,202)(158,203)(159,195)(160,196)(161,197)
(162,216)(163,217)(164,218);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s2*s3*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*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 >;

to this polytope