Polytope of Type {6,72,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,72,2}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15813)
Rank : 4
Schlafli Type : {6,72,2}
Number of vertices, edges, etc : 6, 216, 72, 2
Order of s₀s₁s₂s₃ : 72
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 : {6,36,2}*864a
   3-fold quotients : {2,72,2}*576, {6,24,2}*576a
   4-fold quotients : {6,18,2}*432a
   6-fold quotients : {2,36,2}*288, {6,12,2}*288a
   9-fold quotients : {2,24,2}*192, {6,8,2}*192
   12-fold quotients : {2,18,2}*144, {6,6,2}*144a
   18-fold quotients : {2,12,2}*96, {6,4,2}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,6,2}*48, {6,2,2}*48
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {2,3,2}*24, {3,2,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

to this polytope