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