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