Polytope of Type {24,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,18}*864a
Also Known As : {24,18|2}. if this polytope has another name.
Group : SmallGroup(864,781)
Rank : 3
Schlafli Type : {24,18}
Number of vertices, edges, etc : 24, 216, 18
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {24,18,2} of size 1728
Vertex Figure Of :
   {2,24,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,18}*432a
   3-fold quotients : {8,18}*288, {24,6}*288a
   4-fold quotients : {6,18}*216a
   6-fold quotients : {4,18}*144a, {12,6}*144a
   9-fold quotients : {24,2}*96, {8,6}*96
   12-fold quotients : {2,18}*72, {6,6}*72a
   18-fold quotients : {12,2}*48, {4,6}*48a
   24-fold quotients : {2,9}*36
   27-fold quotients : {8,2}*32
   36-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {48,18}*1728a, {24,36}*1728c
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s0*s1*s2*s1*s0*s1*s2*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, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope