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