Polytope of Type {2,24,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,18}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15830)
Rank : 4
Schlafli Type : {2,24,18}
Number of vertices, edges, etc : 2, 24, 216, 18
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 : {2,12,18}*864a
   3-fold quotients : {2,8,18}*576, {2,24,6}*576a
   4-fold quotients : {2,6,18}*432a
   6-fold quotients : {2,4,18}*288a, {2,12,6}*288a
   9-fold quotients : {2,24,2}*192, {2,8,6}*192
   12-fold quotients : {2,2,18}*144, {2,6,6}*144a
   18-fold quotients : {2,12,2}*96, {2,4,6}*96a
   24-fold quotients : {2,2,9}*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 := ( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)( 18, 27)( 19, 28)
( 20, 29)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)( 44, 53)( 45, 54)
( 46, 55)( 47, 56)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66,102)( 67,103)( 68,104)( 69,105)( 70,106)
( 71,107)( 72,108)( 73,109)( 74,110)( 75, 93)( 76, 94)( 77, 95)( 78, 96)
( 79, 97)( 80, 98)( 81, 99)( 82,100)( 83,101)(111,165)(112,166)(113,167)
(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,183)(121,184)
(122,185)(123,186)(124,187)(125,188)(126,189)(127,190)(128,191)(129,174)
(130,175)(131,176)(132,177)(133,178)(134,179)(135,180)(136,181)(137,182)
(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)
(146,200)(147,210)(148,211)(149,212)(150,213)(151,214)(152,215)(153,216)
(154,217)(155,218)(156,201)(157,202)(158,203)(159,204)(160,205)(161,206)
(162,207)(163,208)(164,209);;
s2 := (  3,120)(  4,122)(  5,121)(  6,128)(  7,127)(  8,126)(  9,125)( 10,124)
( 11,123)( 12,111)( 13,113)( 14,112)( 15,119)( 16,118)( 17,117)( 18,116)
( 19,115)( 20,114)( 21,129)( 22,131)( 23,130)( 24,137)( 25,136)( 26,135)
( 27,134)( 28,133)( 29,132)( 30,147)( 31,149)( 32,148)( 33,155)( 34,154)
( 35,153)( 36,152)( 37,151)( 38,150)( 39,138)( 40,140)( 41,139)( 42,146)
( 43,145)( 44,144)( 45,143)( 46,142)( 47,141)( 48,156)( 49,158)( 50,157)
( 51,164)( 52,163)( 53,162)( 54,161)( 55,160)( 56,159)( 57,201)( 58,203)
( 59,202)( 60,209)( 61,208)( 62,207)( 63,206)( 64,205)( 65,204)( 66,192)
( 67,194)( 68,193)( 69,200)( 70,199)( 71,198)( 72,197)( 73,196)( 74,195)
( 75,210)( 76,212)( 77,211)( 78,218)( 79,217)( 80,216)( 81,215)( 82,214)
( 83,213)( 84,174)( 85,176)( 86,175)( 87,182)( 88,181)( 89,180)( 90,179)
( 91,178)( 92,177)( 93,165)( 94,167)( 95,166)( 96,173)( 97,172)( 98,171)
( 99,170)(100,169)(101,168)(102,183)(103,185)(104,184)(105,191)(106,190)
(107,189)(108,188)(109,187)(110,186);;
s3 := (  3,  6)(  4,  8)(  5,  7)(  9, 11)( 12, 15)( 13, 17)( 14, 16)( 18, 20)
( 21, 24)( 22, 26)( 23, 25)( 27, 29)( 30, 33)( 31, 35)( 32, 34)( 36, 38)
( 39, 42)( 40, 44)( 41, 43)( 45, 47)( 48, 51)( 49, 53)( 50, 52)( 54, 56)
( 57, 60)( 58, 62)( 59, 61)( 63, 65)( 66, 69)( 67, 71)( 68, 70)( 72, 74)
( 75, 78)( 76, 80)( 77, 79)( 81, 83)( 84, 87)( 85, 89)( 86, 88)( 90, 92)
( 93, 96)( 94, 98)( 95, 97)( 99,101)(102,105)(103,107)(104,106)(108,110)
(111,114)(112,116)(113,115)(117,119)(120,123)(121,125)(122,124)(126,128)
(129,132)(130,134)(131,133)(135,137)(138,141)(139,143)(140,142)(144,146)
(147,150)(148,152)(149,151)(153,155)(156,159)(157,161)(158,160)(162,164)
(165,168)(166,170)(167,169)(171,173)(174,177)(175,179)(176,178)(180,182)
(183,186)(184,188)(185,187)(189,191)(192,195)(193,197)(194,196)(198,200)
(201,204)(202,206)(203,205)(207,209)(210,213)(211,215)(212,214)(216,218);;
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*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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

to this polytope