Polytope of Type {4,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1944a
if this polytope has a name.
Group : SmallGroup(1944,804)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 81, 486, 243
Order of s0s1s2 : 18
Order of s0s1s2s1 : 9
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
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) :
3-fold quotients : {4,4}*648
9-fold quotients : {4,12}*216
27-fold quotients : {4,4}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 5)( 3, 9)( 4, 7)( 10, 19)( 11, 23)( 12, 27)( 13, 25)( 14, 20)
( 15, 24)( 16, 22)( 17, 26)( 18, 21)( 28, 40)( 29, 44)( 30, 39)( 31, 37)
( 32, 41)( 33, 45)( 34, 43)( 35, 38)( 36, 42)( 46, 49)( 47, 53)( 51, 54)
( 55, 79)( 56, 74)( 57, 78)( 58, 76)( 59, 80)( 60, 75)( 61, 73)( 62, 77)
( 63, 81)( 64, 70)( 66, 69)( 68, 71)( 82,169)( 83,164)( 84,168)( 85,166)
( 86,170)( 87,165)( 88,163)( 89,167)( 90,171)( 91,187)( 92,182)( 93,186)
( 94,184)( 95,188)( 96,183)( 97,181)( 98,185)( 99,189)(100,178)(101,173)
(102,177)(103,175)(104,179)(105,174)(106,172)(107,176)(108,180)(109,199)
(110,203)(111,207)(112,205)(113,200)(114,204)(115,202)(116,206)(117,201)
(118,190)(119,194)(120,198)(121,196)(122,191)(123,195)(124,193)(125,197)
(126,192)(127,208)(128,212)(129,216)(130,214)(131,209)(132,213)(133,211)
(134,215)(135,210)(136,238)(137,242)(138,237)(139,235)(140,239)(141,243)
(142,241)(143,236)(144,240)(145,229)(146,233)(147,228)(148,226)(149,230)
(150,234)(151,232)(152,227)(153,231)(154,220)(155,224)(156,219)(157,217)
(158,221)(159,225)(160,223)(161,218)(162,222);;
s1 := ( 2, 3)( 4, 6)( 7, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)( 14, 23)
( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28,186)( 29,185)( 30,184)( 31,188)
( 32,187)( 33,189)( 34,181)( 35,183)( 36,182)( 37,177)( 38,176)( 39,175)
( 40,179)( 41,178)( 42,180)( 43,172)( 44,174)( 45,173)( 46,168)( 47,167)
( 48,166)( 49,170)( 50,169)( 51,171)( 52,163)( 53,165)( 54,164)( 55, 92)
( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)( 63, 97)
( 64, 83)( 65, 82)( 66, 84)( 67, 85)( 68, 87)( 69, 86)( 70, 90)( 71, 89)
( 72, 88)( 73,101)( 74,100)( 75,102)( 76,103)( 77,105)( 78,104)( 79,108)
( 80,107)( 81,106)(109,229)(110,231)(111,230)(112,234)(113,233)(114,232)
(115,227)(116,226)(117,228)(118,220)(119,222)(120,221)(121,225)(122,224)
(123,223)(124,218)(125,217)(126,219)(127,238)(128,240)(129,239)(130,243)
(131,242)(132,241)(133,236)(134,235)(135,237)(136,147)(137,146)(138,145)
(139,149)(140,148)(141,150)(142,151)(143,153)(144,152)(154,156)(157,158)
(161,162)(190,192)(193,194)(197,198)(199,210)(200,209)(201,208)(202,212)
(203,211)(204,213)(205,214)(206,216)(207,215);;
s2 := ( 1, 67)( 2, 66)( 3, 71)( 4, 70)( 5, 69)( 6, 65)( 7, 64)( 8, 72)
( 9, 68)( 10, 58)( 11, 57)( 12, 62)( 13, 61)( 14, 60)( 15, 56)( 16, 55)
( 17, 63)( 18, 59)( 19, 76)( 20, 75)( 21, 80)( 22, 79)( 23, 78)( 24, 74)
( 25, 73)( 26, 81)( 27, 77)( 28, 37)( 29, 45)( 30, 41)( 31, 40)( 32, 39)
( 33, 44)( 34, 43)( 35, 42)( 36, 38)( 47, 54)( 48, 50)( 51, 53)( 82,148)
( 83,147)( 84,152)( 85,151)( 86,150)( 87,146)( 88,145)( 89,153)( 90,149)
( 91,139)( 92,138)( 93,143)( 94,142)( 95,141)( 96,137)( 97,136)( 98,144)
( 99,140)(100,157)(101,156)(102,161)(103,160)(104,159)(105,155)(106,154)
(107,162)(108,158)(109,118)(110,126)(111,122)(112,121)(113,120)(114,125)
(115,124)(116,123)(117,119)(128,135)(129,131)(132,134)(163,229)(164,228)
(165,233)(166,232)(167,231)(168,227)(169,226)(170,234)(171,230)(172,220)
(173,219)(174,224)(175,223)(176,222)(177,218)(178,217)(179,225)(180,221)
(181,238)(182,237)(183,242)(184,241)(185,240)(186,236)(187,235)(188,243)
(189,239)(190,199)(191,207)(192,203)(193,202)(194,201)(195,206)(196,205)
(197,204)(198,200)(209,216)(210,212)(213,215);;
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(243)!( 2, 5)( 3, 9)( 4, 7)( 10, 19)( 11, 23)( 12, 27)( 13, 25)
( 14, 20)( 15, 24)( 16, 22)( 17, 26)( 18, 21)( 28, 40)( 29, 44)( 30, 39)
( 31, 37)( 32, 41)( 33, 45)( 34, 43)( 35, 38)( 36, 42)( 46, 49)( 47, 53)
( 51, 54)( 55, 79)( 56, 74)( 57, 78)( 58, 76)( 59, 80)( 60, 75)( 61, 73)
( 62, 77)( 63, 81)( 64, 70)( 66, 69)( 68, 71)( 82,169)( 83,164)( 84,168)
( 85,166)( 86,170)( 87,165)( 88,163)( 89,167)( 90,171)( 91,187)( 92,182)
( 93,186)( 94,184)( 95,188)( 96,183)( 97,181)( 98,185)( 99,189)(100,178)
(101,173)(102,177)(103,175)(104,179)(105,174)(106,172)(107,176)(108,180)
(109,199)(110,203)(111,207)(112,205)(113,200)(114,204)(115,202)(116,206)
(117,201)(118,190)(119,194)(120,198)(121,196)(122,191)(123,195)(124,193)
(125,197)(126,192)(127,208)(128,212)(129,216)(130,214)(131,209)(132,213)
(133,211)(134,215)(135,210)(136,238)(137,242)(138,237)(139,235)(140,239)
(141,243)(142,241)(143,236)(144,240)(145,229)(146,233)(147,228)(148,226)
(149,230)(150,234)(151,232)(152,227)(153,231)(154,220)(155,224)(156,219)
(157,217)(158,221)(159,225)(160,223)(161,218)(162,222);
s1 := Sym(243)!( 2, 3)( 4, 6)( 7, 8)( 10, 19)( 11, 21)( 12, 20)( 13, 24)
( 14, 23)( 15, 22)( 16, 26)( 17, 25)( 18, 27)( 28,186)( 29,185)( 30,184)
( 31,188)( 32,187)( 33,189)( 34,181)( 35,183)( 36,182)( 37,177)( 38,176)
( 39,175)( 40,179)( 41,178)( 42,180)( 43,172)( 44,174)( 45,173)( 46,168)
( 47,167)( 48,166)( 49,170)( 50,169)( 51,171)( 52,163)( 53,165)( 54,164)
( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 99)( 62, 98)
( 63, 97)( 64, 83)( 65, 82)( 66, 84)( 67, 85)( 68, 87)( 69, 86)( 70, 90)
( 71, 89)( 72, 88)( 73,101)( 74,100)( 75,102)( 76,103)( 77,105)( 78,104)
( 79,108)( 80,107)( 81,106)(109,229)(110,231)(111,230)(112,234)(113,233)
(114,232)(115,227)(116,226)(117,228)(118,220)(119,222)(120,221)(121,225)
(122,224)(123,223)(124,218)(125,217)(126,219)(127,238)(128,240)(129,239)
(130,243)(131,242)(132,241)(133,236)(134,235)(135,237)(136,147)(137,146)
(138,145)(139,149)(140,148)(141,150)(142,151)(143,153)(144,152)(154,156)
(157,158)(161,162)(190,192)(193,194)(197,198)(199,210)(200,209)(201,208)
(202,212)(203,211)(204,213)(205,214)(206,216)(207,215);
s2 := Sym(243)!( 1, 67)( 2, 66)( 3, 71)( 4, 70)( 5, 69)( 6, 65)( 7, 64)
( 8, 72)( 9, 68)( 10, 58)( 11, 57)( 12, 62)( 13, 61)( 14, 60)( 15, 56)
( 16, 55)( 17, 63)( 18, 59)( 19, 76)( 20, 75)( 21, 80)( 22, 79)( 23, 78)
( 24, 74)( 25, 73)( 26, 81)( 27, 77)( 28, 37)( 29, 45)( 30, 41)( 31, 40)
( 32, 39)( 33, 44)( 34, 43)( 35, 42)( 36, 38)( 47, 54)( 48, 50)( 51, 53)
( 82,148)( 83,147)( 84,152)( 85,151)( 86,150)( 87,146)( 88,145)( 89,153)
( 90,149)( 91,139)( 92,138)( 93,143)( 94,142)( 95,141)( 96,137)( 97,136)
( 98,144)( 99,140)(100,157)(101,156)(102,161)(103,160)(104,159)(105,155)
(106,154)(107,162)(108,158)(109,118)(110,126)(111,122)(112,121)(113,120)
(114,125)(115,124)(116,123)(117,119)(128,135)(129,131)(132,134)(163,229)
(164,228)(165,233)(166,232)(167,231)(168,227)(169,226)(170,234)(171,230)
(172,220)(173,219)(174,224)(175,223)(176,222)(177,218)(178,217)(179,225)
(180,221)(181,238)(182,237)(183,242)(184,241)(185,240)(186,236)(187,235)
(188,243)(189,239)(190,199)(191,207)(192,203)(193,202)(194,201)(195,206)
(196,205)(197,204)(198,200)(209,216)(210,212)(213,215);
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, s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope