Polytope of Type {24,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,18}*864b
if this polytope has a name.
Group : SmallGroup(864,1141)
Rank : 3
Schlafli Type : {24,18}
Number of vertices, edges, etc : 24, 216, 18
Order of s₀s₁s₂ : 72
Order of s₀s₁s₂s₁ : 6
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}*432b
   3-fold quotients : {8,18}*288, {24,6}*288c
   4-fold quotients : {6,18}*216b
   6-fold quotients : {4,18}*144a, {12,6}*144c
   8-fold quotients : {6,9}*108
   9-fold quotients : {8,6}*96
   12-fold quotients : {2,18}*72, {6,6}*72b
   18-fold quotients : {4,6}*48a
   24-fold quotients : {2,9}*36, {6,3}*36
   27-fold quotients : {8,2}*32
   36-fold quotients : {2,6}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {2,3}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {24,36}*1728b, {48,18}*1728b
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)( 55, 82)( 56, 83)( 57, 84)( 58, 88)( 59, 89)( 60, 90)
( 61, 85)( 62, 86)( 63, 87)( 64, 91)( 65, 92)( 66, 93)( 67, 97)( 68, 98)
( 69, 99)( 70, 94)( 71, 95)( 72, 96)( 73,100)( 74,101)( 75,102)( 76,106)
( 77,107)( 78,108)( 79,103)( 80,104)( 81,105)(109,163)(110,164)(111,165)
(112,169)(113,170)(114,171)(115,166)(116,167)(117,168)(118,172)(119,173)
(120,174)(121,178)(122,179)(123,180)(124,175)(125,176)(126,177)(127,181)
(128,182)(129,183)(130,187)(131,188)(132,189)(133,184)(134,185)(135,186)
(136,190)(137,191)(138,192)(139,196)(140,197)(141,198)(142,193)(143,194)
(144,195)(145,199)(146,200)(147,201)(148,205)(149,206)(150,207)(151,202)
(152,203)(153,204)(154,208)(155,209)(156,210)(157,214)(158,215)(159,216)
(160,211)(161,212)(162,213);;
s1 := (  1,112)(  2,114)(  3,113)(  4,109)(  5,111)(  6,110)(  7,115)(  8,117)
(  9,116)( 10,132)( 11,131)( 12,130)( 13,129)( 14,128)( 15,127)( 16,135)
( 17,134)( 18,133)( 19,123)( 20,122)( 21,121)( 22,120)( 23,119)( 24,118)
( 25,126)( 26,125)( 27,124)( 28,139)( 29,141)( 30,140)( 31,136)( 32,138)
( 33,137)( 34,142)( 35,144)( 36,143)( 37,159)( 38,158)( 39,157)( 40,156)
( 41,155)( 42,154)( 43,162)( 44,161)( 45,160)( 46,150)( 47,149)( 48,148)
( 49,147)( 50,146)( 51,145)( 52,153)( 53,152)( 54,151)( 55,193)( 56,195)
( 57,194)( 58,190)( 59,192)( 60,191)( 61,196)( 62,198)( 63,197)( 64,213)
( 65,212)( 66,211)( 67,210)( 68,209)( 69,208)( 70,216)( 71,215)( 72,214)
( 73,204)( 74,203)( 75,202)( 76,201)( 77,200)( 78,199)( 79,207)( 80,206)
( 81,205)( 82,166)( 83,168)( 84,167)( 85,163)( 86,165)( 87,164)( 88,169)
( 89,171)( 90,170)( 91,186)( 92,185)( 93,184)( 94,183)( 95,182)( 96,181)
( 97,189)( 98,188)( 99,187)(100,177)(101,176)(102,175)(103,174)(104,173)
(105,172)(106,180)(107,179)(108,178);;
s2 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 46, 48)( 49, 54)
( 50, 53)( 51, 52)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(100,102)(103,108)(104,107)(105,106)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(127,129)(130,135)(131,134)
(132,133)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(154,156)(157,162)(158,161)(159,160)(163,172)(164,174)
(165,173)(166,178)(167,180)(168,179)(169,175)(170,177)(171,176)(181,183)
(184,189)(185,188)(186,187)(190,199)(191,201)(192,200)(193,205)(194,207)
(195,206)(196,202)(197,204)(198,203)(208,210)(211,216)(212,215)(213,214);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope