Polytope of Type {4,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*1944
if this polytope has a name.
Group : SmallGroup(1944,805)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 162, 486, 243
Order of s0s1s2 : 12
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
9-fold quotients : {4,6}*216
27-fold quotients : {4,6}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 4, 6)( 7, 8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)( 14, 27)
( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)( 31,189)
( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)( 39,180)
( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)( 47,164)
( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)( 55, 91)
( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)( 63, 99)
( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)( 71, 83)
( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)( 79,103)
( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)(114,229)
(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)(122,225)
(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)(130,236)
(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)(138,150)
(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)(157,159)
(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)(202,216)
(203,215)(204,214)(205,209)(206,208)(207,210);;
s1 := ( 2, 9)( 3, 5)( 6, 8)( 10, 19)( 11, 27)( 12, 23)( 13, 22)( 14, 21)
( 15, 26)( 16, 25)( 17, 24)( 18, 20)( 28, 44)( 29, 40)( 30, 39)( 31, 38)
( 32, 43)( 33, 42)( 34, 41)( 35, 37)( 36, 45)( 46, 53)( 47, 49)( 50, 52)
( 55, 80)( 56, 76)( 57, 75)( 58, 74)( 59, 79)( 60, 78)( 61, 77)( 62, 73)
( 63, 81)( 64, 71)( 65, 67)( 68, 70)( 82,163)( 83,171)( 84,167)( 85,166)
( 86,165)( 87,170)( 88,169)( 89,168)( 90,164)( 91,181)( 92,189)( 93,185)
( 94,184)( 95,183)( 96,188)( 97,187)( 98,186)( 99,182)(100,172)(101,180)
(102,176)(103,175)(104,174)(105,179)(106,178)(107,177)(108,173)(109,206)
(110,202)(111,201)(112,200)(113,205)(114,204)(115,203)(116,199)(117,207)
(118,197)(119,193)(120,192)(121,191)(122,196)(123,195)(124,194)(125,190)
(126,198)(127,215)(128,211)(129,210)(130,209)(131,214)(132,213)(133,212)
(134,208)(135,216)(136,242)(137,238)(138,237)(139,236)(140,241)(141,240)
(142,239)(143,235)(144,243)(145,233)(146,229)(147,228)(148,227)(149,232)
(150,231)(151,230)(152,226)(153,234)(154,224)(155,220)(156,219)(157,218)
(158,223)(159,222)(160,221)(161,217)(162,225);;
s2 := ( 1,191)( 2,190)( 3,192)( 4,197)( 5,196)( 6,198)( 7,194)( 8,193)
( 9,195)( 10,206)( 11,205)( 12,207)( 13,203)( 14,202)( 15,204)( 16,200)
( 17,199)( 18,201)( 19,212)( 20,211)( 21,213)( 22,209)( 23,208)( 24,210)
( 25,215)( 26,214)( 27,216)( 28,164)( 29,163)( 30,165)( 31,170)( 32,169)
( 33,171)( 34,167)( 35,166)( 36,168)( 37,179)( 38,178)( 39,180)( 40,176)
( 41,175)( 42,177)( 43,173)( 44,172)( 45,174)( 46,185)( 47,184)( 48,186)
( 49,182)( 50,181)( 51,183)( 52,188)( 53,187)( 54,189)( 55,218)( 56,217)
( 57,219)( 58,224)( 59,223)( 60,225)( 61,221)( 62,220)( 63,222)( 64,233)
( 65,232)( 66,234)( 67,230)( 68,229)( 69,231)( 70,227)( 71,226)( 72,228)
( 73,239)( 74,238)( 75,240)( 76,236)( 77,235)( 78,237)( 79,242)( 80,241)
( 81,243)( 82,110)( 83,109)( 84,111)( 85,116)( 86,115)( 87,117)( 88,113)
( 89,112)( 90,114)( 91,125)( 92,124)( 93,126)( 94,122)( 95,121)( 96,123)
( 97,119)( 98,118)( 99,120)(100,131)(101,130)(102,132)(103,128)(104,127)
(105,129)(106,134)(107,133)(108,135)(136,137)(139,143)(140,142)(141,144)
(145,152)(146,151)(147,153)(148,149)(154,158)(155,157)(156,159)(160,161);;
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*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 3)( 4, 6)( 7, 8)( 10, 23)( 11, 22)( 12, 24)( 13, 25)
( 14, 27)( 15, 26)( 16, 21)( 17, 20)( 18, 19)( 28,184)( 29,186)( 30,185)
( 31,189)( 32,188)( 33,187)( 34,182)( 35,181)( 36,183)( 37,179)( 38,178)
( 39,180)( 40,172)( 41,174)( 42,173)( 43,177)( 44,176)( 45,175)( 46,165)
( 47,164)( 48,163)( 49,167)( 50,166)( 51,168)( 52,169)( 53,171)( 54,170)
( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)
( 63, 99)( 64, 86)( 65, 85)( 66, 87)( 67, 88)( 68, 90)( 69, 89)( 70, 84)
( 71, 83)( 72, 82)( 73,108)( 74,107)( 75,106)( 76,101)( 77,100)( 78,102)
( 79,103)( 80,105)( 81,104)(109,226)(110,228)(111,227)(112,231)(113,230)
(114,229)(115,233)(116,232)(117,234)(118,221)(119,220)(120,222)(121,223)
(122,225)(123,224)(124,219)(125,218)(126,217)(127,243)(128,242)(129,241)
(130,236)(131,235)(132,237)(133,238)(134,240)(135,239)(136,149)(137,148)
(138,150)(139,151)(140,153)(141,152)(142,147)(143,146)(144,145)(155,156)
(157,159)(160,161)(190,192)(193,194)(197,198)(199,211)(200,213)(201,212)
(202,216)(203,215)(204,214)(205,209)(206,208)(207,210);
s1 := Sym(243)!( 2, 9)( 3, 5)( 6, 8)( 10, 19)( 11, 27)( 12, 23)( 13, 22)
( 14, 21)( 15, 26)( 16, 25)( 17, 24)( 18, 20)( 28, 44)( 29, 40)( 30, 39)
( 31, 38)( 32, 43)( 33, 42)( 34, 41)( 35, 37)( 36, 45)( 46, 53)( 47, 49)
( 50, 52)( 55, 80)( 56, 76)( 57, 75)( 58, 74)( 59, 79)( 60, 78)( 61, 77)
( 62, 73)( 63, 81)( 64, 71)( 65, 67)( 68, 70)( 82,163)( 83,171)( 84,167)
( 85,166)( 86,165)( 87,170)( 88,169)( 89,168)( 90,164)( 91,181)( 92,189)
( 93,185)( 94,184)( 95,183)( 96,188)( 97,187)( 98,186)( 99,182)(100,172)
(101,180)(102,176)(103,175)(104,174)(105,179)(106,178)(107,177)(108,173)
(109,206)(110,202)(111,201)(112,200)(113,205)(114,204)(115,203)(116,199)
(117,207)(118,197)(119,193)(120,192)(121,191)(122,196)(123,195)(124,194)
(125,190)(126,198)(127,215)(128,211)(129,210)(130,209)(131,214)(132,213)
(133,212)(134,208)(135,216)(136,242)(137,238)(138,237)(139,236)(140,241)
(141,240)(142,239)(143,235)(144,243)(145,233)(146,229)(147,228)(148,227)
(149,232)(150,231)(151,230)(152,226)(153,234)(154,224)(155,220)(156,219)
(157,218)(158,223)(159,222)(160,221)(161,217)(162,225);
s2 := Sym(243)!( 1,191)( 2,190)( 3,192)( 4,197)( 5,196)( 6,198)( 7,194)
( 8,193)( 9,195)( 10,206)( 11,205)( 12,207)( 13,203)( 14,202)( 15,204)
( 16,200)( 17,199)( 18,201)( 19,212)( 20,211)( 21,213)( 22,209)( 23,208)
( 24,210)( 25,215)( 26,214)( 27,216)( 28,164)( 29,163)( 30,165)( 31,170)
( 32,169)( 33,171)( 34,167)( 35,166)( 36,168)( 37,179)( 38,178)( 39,180)
( 40,176)( 41,175)( 42,177)( 43,173)( 44,172)( 45,174)( 46,185)( 47,184)
( 48,186)( 49,182)( 50,181)( 51,183)( 52,188)( 53,187)( 54,189)( 55,218)
( 56,217)( 57,219)( 58,224)( 59,223)( 60,225)( 61,221)( 62,220)( 63,222)
( 64,233)( 65,232)( 66,234)( 67,230)( 68,229)( 69,231)( 70,227)( 71,226)
( 72,228)( 73,239)( 74,238)( 75,240)( 76,236)( 77,235)( 78,237)( 79,242)
( 80,241)( 81,243)( 82,110)( 83,109)( 84,111)( 85,116)( 86,115)( 87,117)
( 88,113)( 89,112)( 90,114)( 91,125)( 92,124)( 93,126)( 94,122)( 95,121)
( 96,123)( 97,119)( 98,118)( 99,120)(100,131)(101,130)(102,132)(103,128)
(104,127)(105,129)(106,134)(107,133)(108,135)(136,137)(139,143)(140,142)
(141,144)(145,152)(146,151)(147,153)(148,149)(154,158)(155,157)(156,159)
(160,161);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s2*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1 >;
References : None.
to this polytope