Polytope of Type {2,4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,12}*1728d
if this polytope has a name.
Group : SmallGroup(1728,46611)
Rank : 4
Schlafli Type : {2,4,12}
Number of vertices, edges, etc : 2, 36, 216, 108
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 : {2,4,12}*864b
   3-fold quotients : {2,4,4}*576
   6-fold quotients : {2,4,4}*288
   9-fold quotients : {2,4,12}*192a
   12-fold quotients : {2,4,4}*144
   18-fold quotients : {2,2,12}*96, {2,4,6}*96a
   27-fold quotients : {2,4,4}*64
   36-fold quotients : {2,2,6}*48
   54-fold quotients : {2,2,4}*32, {2,4,2}*32
   72-fold quotients : {2,2,3}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

to this polytope