Polytope of Type {4,12,2}

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

to this polytope