Polytope of Type {2,72,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,72,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15813)
Rank : 4
Schlafli Type : {2,72,6}
Number of vertices, edges, etc : 2, 72, 216, 6
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 : {2,36,6}*864a
   3-fold quotients : {2,72,2}*576, {2,24,6}*576a
   4-fold quotients : {2,18,6}*432a
   6-fold quotients : {2,36,2}*288, {2,12,6}*288a
   9-fold quotients : {2,24,2}*192, {2,8,6}*192
   12-fold quotients : {2,18,2}*144, {2,6,6}*144a
   18-fold quotients : {2,12,2}*96, {2,4,6}*96a
   24-fold quotients : {2,9,2}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,4,2}*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 := (1,2);;
s1 := (  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)( 16, 25)
( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)( 39, 50)
( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)( 47, 54)
( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)( 64, 92)
( 65, 91)( 66,104)( 67,103)( 68,102)( 69,107)( 70,106)( 71,105)( 72,110)
( 73,109)( 74,108)( 75, 95)( 76, 94)( 77, 93)( 78, 98)( 79, 97)( 80, 96)
( 81,101)( 82,100)( 83, 99)(111,165)(112,167)(113,166)(114,168)(115,170)
(116,169)(117,171)(118,173)(119,172)(120,185)(121,184)(122,183)(123,188)
(124,187)(125,186)(126,191)(127,190)(128,189)(129,176)(130,175)(131,174)
(132,179)(133,178)(134,177)(135,182)(136,181)(137,180)(138,192)(139,194)
(140,193)(141,195)(142,197)(143,196)(144,198)(145,200)(146,199)(147,212)
(148,211)(149,210)(150,215)(151,214)(152,213)(153,218)(154,217)(155,216)
(156,203)(157,202)(158,201)(159,206)(160,205)(161,204)(162,209)(163,208)
(164,207);;
s2 := (  3,120)(  4,122)(  5,121)(  6,126)(  7,128)(  8,127)(  9,123)( 10,125)
( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)( 18,114)
( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)( 34,155)
( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)( 42,144)
( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)( 50,156)
( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,201)( 58,203)
( 59,202)( 60,207)( 61,209)( 62,208)( 63,204)( 64,206)( 65,205)( 66,192)
( 67,194)( 68,193)( 69,198)( 70,200)( 71,199)( 72,195)( 73,197)( 74,196)
( 75,212)( 76,211)( 77,210)( 78,218)( 79,217)( 80,216)( 81,215)( 82,214)
( 83,213)( 84,174)( 85,176)( 86,175)( 87,180)( 88,182)( 89,181)( 90,177)
( 91,179)( 92,178)( 93,165)( 94,167)( 95,166)( 96,171)( 97,173)( 98,172)
( 99,168)(100,170)(101,169)(102,185)(103,184)(104,183)(105,191)(106,190)
(107,189)(108,188)(109,187)(110,186);;
s3 := (  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)( 22, 25)
( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)( 48, 51)
( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)( 68, 71)
( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)( 94, 97)
( 95, 98)(102,105)(103,106)(104,107)(111,114)(112,115)(113,116)(120,123)
(121,124)(122,125)(129,132)(130,133)(131,134)(138,141)(139,142)(140,143)
(147,150)(148,151)(149,152)(156,159)(157,160)(158,161)(165,168)(166,169)
(167,170)(174,177)(175,178)(176,179)(183,186)(184,187)(185,188)(192,195)
(193,196)(194,197)(201,204)(202,205)(203,206)(210,213)(211,214)(212,215);;
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*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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)!(1,2);
s1 := Sym(218)!(  4,  5)(  7,  8)( 10, 11)( 12, 23)( 13, 22)( 14, 21)( 15, 26)
( 16, 25)( 17, 24)( 18, 29)( 19, 28)( 20, 27)( 31, 32)( 34, 35)( 37, 38)
( 39, 50)( 40, 49)( 41, 48)( 42, 53)( 43, 52)( 44, 51)( 45, 56)( 46, 55)
( 47, 54)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)
( 64, 92)( 65, 91)( 66,104)( 67,103)( 68,102)( 69,107)( 70,106)( 71,105)
( 72,110)( 73,109)( 74,108)( 75, 95)( 76, 94)( 77, 93)( 78, 98)( 79, 97)
( 80, 96)( 81,101)( 82,100)( 83, 99)(111,165)(112,167)(113,166)(114,168)
(115,170)(116,169)(117,171)(118,173)(119,172)(120,185)(121,184)(122,183)
(123,188)(124,187)(125,186)(126,191)(127,190)(128,189)(129,176)(130,175)
(131,174)(132,179)(133,178)(134,177)(135,182)(136,181)(137,180)(138,192)
(139,194)(140,193)(141,195)(142,197)(143,196)(144,198)(145,200)(146,199)
(147,212)(148,211)(149,210)(150,215)(151,214)(152,213)(153,218)(154,217)
(155,216)(156,203)(157,202)(158,201)(159,206)(160,205)(161,204)(162,209)
(163,208)(164,207);
s2 := Sym(218)!(  3,120)(  4,122)(  5,121)(  6,126)(  7,128)(  8,127)(  9,123)
( 10,125)( 11,124)( 12,111)( 13,113)( 14,112)( 15,117)( 16,119)( 17,118)
( 18,114)( 19,116)( 20,115)( 21,131)( 22,130)( 23,129)( 24,137)( 25,136)
( 26,135)( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,153)
( 34,155)( 35,154)( 36,150)( 37,152)( 38,151)( 39,138)( 40,140)( 41,139)
( 42,144)( 43,146)( 44,145)( 45,141)( 46,143)( 47,142)( 48,158)( 49,157)
( 50,156)( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,201)
( 58,203)( 59,202)( 60,207)( 61,209)( 62,208)( 63,204)( 64,206)( 65,205)
( 66,192)( 67,194)( 68,193)( 69,198)( 70,200)( 71,199)( 72,195)( 73,197)
( 74,196)( 75,212)( 76,211)( 77,210)( 78,218)( 79,217)( 80,216)( 81,215)
( 82,214)( 83,213)( 84,174)( 85,176)( 86,175)( 87,180)( 88,182)( 89,181)
( 90,177)( 91,179)( 92,178)( 93,165)( 94,167)( 95,166)( 96,171)( 97,173)
( 98,172)( 99,168)(100,170)(101,169)(102,185)(103,184)(104,183)(105,191)
(106,190)(107,189)(108,188)(109,187)(110,186);
s3 := Sym(218)!(  3,  6)(  4,  7)(  5,  8)( 12, 15)( 13, 16)( 14, 17)( 21, 24)
( 22, 25)( 23, 26)( 30, 33)( 31, 34)( 32, 35)( 39, 42)( 40, 43)( 41, 44)
( 48, 51)( 49, 52)( 50, 53)( 57, 60)( 58, 61)( 59, 62)( 66, 69)( 67, 70)
( 68, 71)( 75, 78)( 76, 79)( 77, 80)( 84, 87)( 85, 88)( 86, 89)( 93, 96)
( 94, 97)( 95, 98)(102,105)(103,106)(104,107)(111,114)(112,115)(113,116)
(120,123)(121,124)(122,125)(129,132)(130,133)(131,134)(138,141)(139,142)
(140,143)(147,150)(148,151)(149,152)(156,159)(157,160)(158,161)(165,168)
(166,169)(167,170)(174,177)(175,178)(176,179)(183,186)(184,187)(185,188)
(192,195)(193,196)(194,197)(201,204)(202,205)(203,206)(210,213)(211,214)
(212,215);
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*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
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