Polytope of Type {4,12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12,6}*1728b
if this polytope has a name.
Group : SmallGroup(1728,14771)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 72, 108, 18
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 : {2,12,6}*864b, {4,6,6}*864b
   3-fold quotients : {4,12,6}*576a
   4-fold quotients : {2,6,6}*432b
   6-fold quotients : {2,12,6}*288a, {4,6,6}*288a
   8-fold quotients : {2,6,6}*216
   9-fold quotients : {4,12,2}*192a, {4,4,6}*192
   12-fold quotients : {2,6,6}*144a
   18-fold quotients : {2,12,2}*96, {2,4,6}*96a, {4,2,6}*96, {4,6,2}*96a
   27-fold quotients : {4,4,2}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,4,2}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope