Polytope of Type {2,72,6}

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