Polytope of Type {2,9,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,9,18}*1944j
if this polytope has a name.
Group : SmallGroup(1944,952)
Rank : 4
Schlafli Type : {2,9,18}
Number of vertices, edges, etc : 2, 27, 243, 54
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {2,9,6}*648b, {2,3,18}*648
9-fold quotients : {2,3,6}*216
27-fold quotients : {2,3,6}*72
81-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := ( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)( 16, 25)
( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)( 33, 62)
( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)( 41, 76)
( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)( 49, 68)
( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)( 84,183)
( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)( 92,191)
( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)(100,180)
(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)(108,172)
(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)(116,240)
(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)(124,232)
(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)(132,224)
(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)(140,211)
(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)(148,203)
(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)(156,192)
(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)(164,200);;
s2 := ( 3,195)( 4,197)( 5,196)( 6,198)( 7,200)( 8,199)( 9,192)( 10,194)
( 11,193)( 12,213)( 13,215)( 14,214)( 15,216)( 16,218)( 17,217)( 18,210)
( 19,212)( 20,211)( 21,204)( 22,206)( 23,205)( 24,207)( 25,209)( 26,208)
( 27,201)( 28,203)( 29,202)( 30,171)( 31,173)( 32,172)( 33,165)( 34,167)
( 35,166)( 36,168)( 37,170)( 38,169)( 39,189)( 40,191)( 41,190)( 42,183)
( 43,185)( 44,184)( 45,186)( 46,188)( 47,187)( 48,180)( 49,182)( 50,181)
( 51,174)( 52,176)( 53,175)( 54,177)( 55,179)( 56,178)( 57,221)( 58,220)
( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)( 66,239)
( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)( 74,243)
( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)( 82,235)
( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)( 90,111)
( 91,113)( 92,112)( 93,132)( 94,134)( 95,133)( 96,135)( 97,137)( 98,136)
( 99,129)(100,131)(101,130)(102,123)(103,125)(104,124)(105,126)(106,128)
(107,127)(108,120)(109,122)(110,121)(138,140)(141,143)(144,146)(147,158)
(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)(155,162);;
s3 := ( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 27)
( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)( 45, 51)
( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)( 67, 77)
( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)( 84,183)
( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)( 92,187)
( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)(100,179)
(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)(108,168)
(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)(116,217)
(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)(124,209)
(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)(132,198)
(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)(140,238)
(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)(148,230)
(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)(156,219)
(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)(164,223);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(245)!(1,2);
s1 := Sym(245)!( 4, 5)( 6, 8)( 9, 10)( 12, 21)( 13, 23)( 14, 22)( 15, 26)
( 16, 25)( 17, 24)( 18, 28)( 19, 27)( 20, 29)( 30, 57)( 31, 59)( 32, 58)
( 33, 62)( 34, 61)( 35, 60)( 36, 64)( 37, 63)( 38, 65)( 39, 75)( 40, 77)
( 41, 76)( 42, 80)( 43, 79)( 44, 78)( 45, 82)( 46, 81)( 47, 83)( 48, 66)
( 49, 68)( 50, 67)( 51, 71)( 52, 70)( 53, 69)( 54, 73)( 55, 72)( 56, 74)
( 84,183)( 85,185)( 86,184)( 87,188)( 88,187)( 89,186)( 90,190)( 91,189)
( 92,191)( 93,174)( 94,176)( 95,175)( 96,179)( 97,178)( 98,177)( 99,181)
(100,180)(101,182)(102,165)(103,167)(104,166)(105,170)(106,169)(107,168)
(108,172)(109,171)(110,173)(111,237)(112,239)(113,238)(114,242)(115,241)
(116,240)(117,244)(118,243)(119,245)(120,228)(121,230)(122,229)(123,233)
(124,232)(125,231)(126,235)(127,234)(128,236)(129,219)(130,221)(131,220)
(132,224)(133,223)(134,222)(135,226)(136,225)(137,227)(138,210)(139,212)
(140,211)(141,215)(142,214)(143,213)(144,217)(145,216)(146,218)(147,201)
(148,203)(149,202)(150,206)(151,205)(152,204)(153,208)(154,207)(155,209)
(156,192)(157,194)(158,193)(159,197)(160,196)(161,195)(162,199)(163,198)
(164,200);
s2 := Sym(245)!( 3,195)( 4,197)( 5,196)( 6,198)( 7,200)( 8,199)( 9,192)
( 10,194)( 11,193)( 12,213)( 13,215)( 14,214)( 15,216)( 16,218)( 17,217)
( 18,210)( 19,212)( 20,211)( 21,204)( 22,206)( 23,205)( 24,207)( 25,209)
( 26,208)( 27,201)( 28,203)( 29,202)( 30,171)( 31,173)( 32,172)( 33,165)
( 34,167)( 35,166)( 36,168)( 37,170)( 38,169)( 39,189)( 40,191)( 41,190)
( 42,183)( 43,185)( 44,184)( 45,186)( 46,188)( 47,187)( 48,180)( 49,182)
( 50,181)( 51,174)( 52,176)( 53,175)( 54,177)( 55,179)( 56,178)( 57,221)
( 58,220)( 59,219)( 60,224)( 61,223)( 62,222)( 63,227)( 64,226)( 65,225)
( 66,239)( 67,238)( 68,237)( 69,242)( 70,241)( 71,240)( 72,245)( 73,244)
( 74,243)( 75,230)( 76,229)( 77,228)( 78,233)( 79,232)( 80,231)( 81,236)
( 82,235)( 83,234)( 84,114)( 85,116)( 86,115)( 87,117)( 88,119)( 89,118)
( 90,111)( 91,113)( 92,112)( 93,132)( 94,134)( 95,133)( 96,135)( 97,137)
( 98,136)( 99,129)(100,131)(101,130)(102,123)(103,125)(104,124)(105,126)
(106,128)(107,127)(108,120)(109,122)(110,121)(138,140)(141,143)(144,146)
(147,158)(148,157)(149,156)(150,161)(151,160)(152,159)(153,164)(154,163)
(155,162);
s3 := Sym(245)!( 4, 5)( 6, 9)( 7, 11)( 8, 10)( 12, 21)( 13, 23)( 14, 22)
( 15, 27)( 16, 29)( 17, 28)( 18, 24)( 19, 26)( 20, 25)( 31, 32)( 33, 36)
( 34, 38)( 35, 37)( 39, 48)( 40, 50)( 41, 49)( 42, 54)( 43, 56)( 44, 55)
( 45, 51)( 46, 53)( 47, 52)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 66, 75)
( 67, 77)( 68, 76)( 69, 81)( 70, 83)( 71, 82)( 72, 78)( 73, 80)( 74, 79)
( 84,183)( 85,185)( 86,184)( 87,189)( 88,191)( 89,190)( 90,186)( 91,188)
( 92,187)( 93,174)( 94,176)( 95,175)( 96,180)( 97,182)( 98,181)( 99,177)
(100,179)(101,178)(102,165)(103,167)(104,166)(105,171)(106,173)(107,172)
(108,168)(109,170)(110,169)(111,210)(112,212)(113,211)(114,216)(115,218)
(116,217)(117,213)(118,215)(119,214)(120,201)(121,203)(122,202)(123,207)
(124,209)(125,208)(126,204)(127,206)(128,205)(129,192)(130,194)(131,193)
(132,198)(133,200)(134,199)(135,195)(136,197)(137,196)(138,237)(139,239)
(140,238)(141,243)(142,245)(143,244)(144,240)(145,242)(146,241)(147,228)
(148,230)(149,229)(150,234)(151,236)(152,235)(153,231)(154,233)(155,232)
(156,219)(157,221)(158,220)(159,225)(160,227)(161,226)(162,222)(163,224)
(164,223);
poly := sub<Sym(245)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s3*s1*s2*s1*s2*s1*s2*s3*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2 >;
to this polytope