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