Polytope of Type {18,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,4}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {18,4}
Number of vertices, edges, etc : 243, 486, 54
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {18,4}*648
9-fold quotients : {6,4}*216
27-fold quotients : {6,4}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 78)( 29, 77)( 30, 76)( 31, 75)
( 32, 74)( 33, 73)( 34, 81)( 35, 80)( 36, 79)( 37, 60)( 38, 59)( 39, 58)
( 40, 57)( 41, 56)( 42, 55)( 43, 63)( 44, 62)( 45, 61)( 46, 69)( 47, 68)
( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)( 54, 70)( 82,169)
( 83,171)( 84,170)( 85,166)( 86,168)( 87,167)( 88,163)( 89,165)( 90,164)
( 91,178)( 92,180)( 93,179)( 94,175)( 95,177)( 96,176)( 97,172)( 98,174)
( 99,173)(100,187)(101,189)(102,188)(103,184)(104,186)(105,185)(106,181)
(107,183)(108,182)(109,237)(110,236)(111,235)(112,243)(113,242)(114,241)
(115,240)(116,239)(117,238)(118,219)(119,218)(120,217)(121,225)(122,224)
(123,223)(124,222)(125,221)(126,220)(127,228)(128,227)(129,226)(130,234)
(131,233)(132,232)(133,231)(134,230)(135,229)(136,201)(137,200)(138,199)
(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,210)(146,209)
(147,208)(148,216)(149,215)(150,214)(151,213)(152,212)(153,211)(154,192)
(155,191)(156,190)(157,198)(158,197)(159,196)(160,195)(161,194)(162,193);;
s1 := ( 1, 82)( 2, 86)( 3, 90)( 4, 88)( 5, 83)( 6, 87)( 7, 85)( 8, 89)
( 9, 84)( 10,100)( 11,104)( 12,108)( 13,106)( 14,101)( 15,105)( 16,103)
( 17,107)( 18,102)( 19, 91)( 20, 95)( 21, 99)( 22, 97)( 23, 92)( 24, 96)
( 25, 94)( 26, 98)( 27, 93)( 28,130)( 29,134)( 30,129)( 31,127)( 32,131)
( 33,135)( 34,133)( 35,128)( 36,132)( 37,121)( 38,125)( 39,120)( 40,118)
( 41,122)( 42,126)( 43,124)( 44,119)( 45,123)( 46,112)( 47,116)( 48,111)
( 49,109)( 50,113)( 51,117)( 52,115)( 53,110)( 54,114)( 55,151)( 56,146)
( 57,150)( 58,148)( 59,152)( 60,147)( 61,145)( 62,149)( 63,153)( 64,142)
( 65,137)( 66,141)( 67,139)( 68,143)( 69,138)( 70,136)( 71,140)( 72,144)
( 73,160)( 74,155)( 75,159)( 76,157)( 77,161)( 78,156)( 79,154)( 80,158)
( 81,162)(163,169)(165,168)(167,170)(172,187)(173,182)(174,186)(175,184)
(176,188)(177,183)(178,181)(179,185)(180,189)(190,208)(191,212)(192,216)
(193,214)(194,209)(195,213)(196,211)(197,215)(198,210)(200,203)(201,207)
(202,205)(217,229)(218,233)(219,228)(220,226)(221,230)(222,234)(223,232)
(224,227)(225,231)(235,238)(236,242)(240,243);;
s2 := ( 1, 10)( 2, 11)( 3, 12)( 4, 17)( 5, 18)( 6, 16)( 7, 15)( 8, 13)
( 9, 14)( 22, 26)( 23, 27)( 24, 25)( 28, 86)( 29, 87)( 30, 85)( 31, 84)
( 32, 82)( 33, 83)( 34, 88)( 35, 89)( 36, 90)( 37,104)( 38,105)( 39,103)
( 40,102)( 41,100)( 42,101)( 43,106)( 44,107)( 45,108)( 46, 95)( 47, 96)
( 48, 94)( 49, 93)( 50, 91)( 51, 92)( 52, 97)( 53, 98)( 54, 99)( 55,189)
( 56,187)( 57,188)( 58,184)( 59,185)( 60,186)( 61,182)( 62,183)( 63,181)
( 64,180)( 65,178)( 66,179)( 67,175)( 68,176)( 69,177)( 70,173)( 71,174)
( 72,172)( 73,171)( 74,169)( 75,170)( 76,166)( 77,167)( 78,168)( 79,164)
( 80,165)( 81,163)(109,117)(110,115)(111,116)(118,135)(119,133)(120,134)
(121,130)(122,131)(123,132)(124,128)(125,129)(126,127)(136,190)(137,191)
(138,192)(139,197)(140,198)(141,196)(142,195)(143,193)(144,194)(145,208)
(146,209)(147,210)(148,215)(149,216)(150,214)(151,213)(152,211)(153,212)
(154,199)(155,200)(156,201)(157,206)(158,207)(159,205)(160,204)(161,202)
(162,203)(217,230)(218,231)(219,229)(220,228)(221,226)(222,227)(223,232)
(224,233)(225,234)(235,239)(236,240)(237,238);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 78)( 29, 77)( 30, 76)
( 31, 75)( 32, 74)( 33, 73)( 34, 81)( 35, 80)( 36, 79)( 37, 60)( 38, 59)
( 39, 58)( 40, 57)( 41, 56)( 42, 55)( 43, 63)( 44, 62)( 45, 61)( 46, 69)
( 47, 68)( 48, 67)( 49, 66)( 50, 65)( 51, 64)( 52, 72)( 53, 71)( 54, 70)
( 82,169)( 83,171)( 84,170)( 85,166)( 86,168)( 87,167)( 88,163)( 89,165)
( 90,164)( 91,178)( 92,180)( 93,179)( 94,175)( 95,177)( 96,176)( 97,172)
( 98,174)( 99,173)(100,187)(101,189)(102,188)(103,184)(104,186)(105,185)
(106,181)(107,183)(108,182)(109,237)(110,236)(111,235)(112,243)(113,242)
(114,241)(115,240)(116,239)(117,238)(118,219)(119,218)(120,217)(121,225)
(122,224)(123,223)(124,222)(125,221)(126,220)(127,228)(128,227)(129,226)
(130,234)(131,233)(132,232)(133,231)(134,230)(135,229)(136,201)(137,200)
(138,199)(139,207)(140,206)(141,205)(142,204)(143,203)(144,202)(145,210)
(146,209)(147,208)(148,216)(149,215)(150,214)(151,213)(152,212)(153,211)
(154,192)(155,191)(156,190)(157,198)(158,197)(159,196)(160,195)(161,194)
(162,193);
s1 := Sym(243)!( 1, 82)( 2, 86)( 3, 90)( 4, 88)( 5, 83)( 6, 87)( 7, 85)
( 8, 89)( 9, 84)( 10,100)( 11,104)( 12,108)( 13,106)( 14,101)( 15,105)
( 16,103)( 17,107)( 18,102)( 19, 91)( 20, 95)( 21, 99)( 22, 97)( 23, 92)
( 24, 96)( 25, 94)( 26, 98)( 27, 93)( 28,130)( 29,134)( 30,129)( 31,127)
( 32,131)( 33,135)( 34,133)( 35,128)( 36,132)( 37,121)( 38,125)( 39,120)
( 40,118)( 41,122)( 42,126)( 43,124)( 44,119)( 45,123)( 46,112)( 47,116)
( 48,111)( 49,109)( 50,113)( 51,117)( 52,115)( 53,110)( 54,114)( 55,151)
( 56,146)( 57,150)( 58,148)( 59,152)( 60,147)( 61,145)( 62,149)( 63,153)
( 64,142)( 65,137)( 66,141)( 67,139)( 68,143)( 69,138)( 70,136)( 71,140)
( 72,144)( 73,160)( 74,155)( 75,159)( 76,157)( 77,161)( 78,156)( 79,154)
( 80,158)( 81,162)(163,169)(165,168)(167,170)(172,187)(173,182)(174,186)
(175,184)(176,188)(177,183)(178,181)(179,185)(180,189)(190,208)(191,212)
(192,216)(193,214)(194,209)(195,213)(196,211)(197,215)(198,210)(200,203)
(201,207)(202,205)(217,229)(218,233)(219,228)(220,226)(221,230)(222,234)
(223,232)(224,227)(225,231)(235,238)(236,242)(240,243);
s2 := Sym(243)!( 1, 10)( 2, 11)( 3, 12)( 4, 17)( 5, 18)( 6, 16)( 7, 15)
( 8, 13)( 9, 14)( 22, 26)( 23, 27)( 24, 25)( 28, 86)( 29, 87)( 30, 85)
( 31, 84)( 32, 82)( 33, 83)( 34, 88)( 35, 89)( 36, 90)( 37,104)( 38,105)
( 39,103)( 40,102)( 41,100)( 42,101)( 43,106)( 44,107)( 45,108)( 46, 95)
( 47, 96)( 48, 94)( 49, 93)( 50, 91)( 51, 92)( 52, 97)( 53, 98)( 54, 99)
( 55,189)( 56,187)( 57,188)( 58,184)( 59,185)( 60,186)( 61,182)( 62,183)
( 63,181)( 64,180)( 65,178)( 66,179)( 67,175)( 68,176)( 69,177)( 70,173)
( 71,174)( 72,172)( 73,171)( 74,169)( 75,170)( 76,166)( 77,167)( 78,168)
( 79,164)( 80,165)( 81,163)(109,117)(110,115)(111,116)(118,135)(119,133)
(120,134)(121,130)(122,131)(123,132)(124,128)(125,129)(126,127)(136,190)
(137,191)(138,192)(139,197)(140,198)(141,196)(142,195)(143,193)(144,194)
(145,208)(146,209)(147,210)(148,215)(149,216)(150,214)(151,213)(152,211)
(153,212)(154,199)(155,200)(156,201)(157,206)(158,207)(159,205)(160,204)
(161,202)(162,203)(217,230)(218,231)(219,229)(220,228)(221,226)(222,227)
(223,232)(224,233)(225,234)(235,239)(236,240)(237,238);
poly := sub<Sym(243)|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, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*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*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope