Polytope of Type {18,24}

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

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