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