Finite Group Databasebeta

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#72263267

order12288
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C27×C43

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72263175 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#72209702
commutator, Fitting, cosocle, hypocenter#72263175C3
FrattiniC24×C4C22⋊(C22⋊A4)
socleC2#72219176
LCSG ▷ #72263175
UCSC1 ◁ C4

At least 11777 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has at least 11777 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degreeμ(G) = 256
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 4 generators on 256 points:

(2,11,24)(3,17,62)(4,25,5)(6,40,7)(12,63,33)(13,57,96)(14,74,110)(15,59,107)(16,18,23)(19,70,69)(20,26,115)(21,82,104)(22,41,145)(27,64,58)(28,118,36)(29,112,133)(30,128,38)(31,34,39)(32,116,90)(35,120,113)(37,55,114)(42,141,89)(43,148,48)(44,143,56)(45,49,51)(46,88,147)(47,131,137)(50,53,144)(52,61,54)(60,103,77)(65,168,106)(66,162,208)(67,174,111)(68,164,99)(71,178,201)(72,153,243)(73,117,198)(75,142,83)(76,188,98)(78,185,109)(79,186,108)(80,189,206)(81,157,244)(84,146,101)(85,86,190)(87,93,161)(91,199,159)(92,177,184)(94,167,95)(97,173,218)(100,211,156)(102,151,160)(105,165,240)(119,222,195)(121,226,138)(122,221,140)(123,228,239)(124,135,125)(126,213,236)(127,214,237)(129,232,197)(130,170,163)(132,224,248)(134,227,217)(136,187,220)(139,225,216)(149,219,193)(150,169,241)(152,242,234)(154,205,180)(155,202,166)(158,196,183)(171,252,209)(172,176,200)(175,250,210)(179,223,215)(181,253,246)(182,251,256)(191,233,235)(192,231,229)(194,212,238)(203,245,255)(204,249,254)(207,230,247);(1,49,10,40)(2,82,17,77)(3,107,24,101)(4,125,31,120)(5,137,39,133)(6,7,45,51)(8,143,9,144)(11,145,62,74)(12,173,26,170)(13,183,32,180)(14,15,41,84)(16,151,52,141)(18,185,61,186)(19,202,27,200)(20,208,33,206)(21,22,103,110)(23,142,54,147)(25,148,34,118)(28,29,43,47)(30,225,55,227)(35,36,135,48)(37,236,38,237)(42,108,102,109)(44,56,53,50)(46,78,83,79)(57,232,116,178)(58,184,69,240)(59,60,146,104)(63,188,115,168)(64,199,70,211)(65,66,76,80)(67,230,68,224)(71,72,129,81)(73,235,85,238)(75,160,88,89)(86,251,117,253)(87,254,95,255)(90,243,96,244)(91,92,100,105)(93,223,94,231)(97,98,130,106)(99,234,111,239)(112,113,131,124)(114,221,128,226)(119,204,149,203)(121,127,122,126)(123,175,152,171)(132,210,207,209)(134,140,139,138)(136,182,150,181)(153,154,157,158)(155,156,176,159)(161,222,167,219)(162,163,189,218)(164,250,174,252)(165,166,177,172)(169,194,187,191)(179,195,192,193)(190,241,198,220)(196,197,205,201)(212,256,233,246)(213,217,214,216)(215,245,229,249)(228,247,242,248);(1,34,10,25)(2,70,17,64)(3,96,24,90)(4,5,31,39)(6,112,45,131)(7,124,51,113)(8,114,9,128)(11,115,62,63)(12,13,26,32)(14,165,41,177)(15,172,84,166)(16,167,52,161)(18,164,61,174)(19,20,27,33)(21,196,103,205)(22,201,110,197)(23,198,54,190)(28,135,43,35)(29,36,47,48)(30,38,55,37)(40,118,49,148)(42,233,102,212)(44,214,53,213)(46,215,83,229)(50,216,56,217)(57,58,116,69)(59,162,146,189)(60,218,104,163)(65,129,76,71)(66,72,80,81)(67,85,68,73)(74,168,145,188)(75,228,88,242)(77,211,82,199)(78,249,79,245)(86,87,117,95)(89,247,160,248)(91,130,100,97)(92,98,105,106)(93,111,94,99)(101,244,107,243)(108,256,109,246)(119,132,149,207)(120,137,125,133)(121,134,122,139)(123,150,152,136)(126,140,127,138)(141,219,151,222)(142,241,147,220)(143,226,144,221)(153,176,157,155)(154,156,158,159)(169,192,187,179)(170,183,173,180)(171,181,175,182)(178,184,232,240)(185,252,186,250)(191,193,194,195)(200,208,202,206)(203,210,204,209)(223,234,231,239)(224,238,230,235)(225,236,227,237)(251,255,253,254);(1,61,8,62,10,18,9,11)(2,24,16,23,17,3,52,54)(4,117,55,57,31,86,30,116)(5,95,38,69,39,87,37,58)(6,146,44,75,45,59,53,88)(7,104,56,160,51,60,50,89)(12,27,68,94,26,19,67,93)(13,20,85,99,32,33,73,111)(14,102,78,21,41,42,79,103)(15,109,83,22,84,108,46,110)(25,164,114,63,34,174,128,115)(28,153,122,169,43,157,121,187)(29,154,126,194,47,158,127,191)(35,155,139,192,135,176,134,179)(36,159,138,195,48,156,140,193)(40,186,144,74,49,185,143,145)(64,90,167,190,70,96,161,198)(65,149,175,91,76,119,171,100)(66,203,152,92,80,204,123,105)(71,207,182,97,129,132,181,130)(72,210,150,106,81,209,136,98)(77,101,141,142,82,107,151,147)(112,189,213,242,131,162,214,228)(113,163,217,247,124,218,216,248)(118,252,226,168,148,250,221,188)(120,253,225,232,125,251,227,178)(133,254,236,184,137,255,237,240)(165,233,245,196,177,212,249,205)(166,256,229,201,172,246,215,197)(170,200,230,231,173,202,224,223)(180,206,235,239,183,208,238,234)(199,243,219,241,211,244,222,220)

Polycyclic presentation:

3,2^12|b^2=h,c^2=hk,d^2=m,e^2=m,f^2=m,g^2=m,h^2=m,i^2=m,b^a=chim,c^a=bci,c^b=ci,d^a=e,d^b=dj,d^c=djkl,e^a=dem,e^b=ejklm,e^c=eklm,e^d=em,f^a=g,f^b=fhijkm,f^c=fhjkl,f^d=fm,f^e=fm,g^a=fgm,g^b=ghjkl,g^c=gl,g^e=gm,g^f=gm,h^a=hkm,h^c=hm,h^d=hm,h^g=hm,j^a=klm,j^b=jm,j^f=jm,j^g=jm,k^a=hi,k^b=km,k^c=km,k^e=km,k^f=km,k^g=km,l^a=hijkl,l^e=lm,l^f=lm

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265798

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k", "l", "m");;
gap> coll := SingleCollector(F, [3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.8);;
gap> SetPower(coll, 3, F.8*F.11);;
gap> SetPower(coll, 4, F.13);;
gap> SetPower(coll, 5, F.13);;
gap> SetPower(coll, 6, F.13);;
gap> SetPower(coll, 7, F.13);;
gap> SetPower(coll, 8, F.13);;
gap> SetPower(coll, 9, F.13);;
gap> SetConjugate(coll, 2, 1, F.3*F.8*F.9*F.13);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10*F.11*F.12);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.13);;
gap> SetConjugate(coll, 5, 2, F.5*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 3, F.5*F.11*F.12*F.13);;
gap> SetConjugate(coll, 5, 4, F.5*F.13);;
gap> SetConjugate(coll, 6, 1, F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10*F.11*F.13);;
gap> SetConjugate(coll, 6, 3, F.6*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.13);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.13);;
gap> SetConjugate(coll, 7, 2, F.7*F.8*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.12);;
gap> SetConjugate(coll, 7, 5, F.7*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.13);;
gap> SetConjugate(coll, 8, 1, F.8*F.11*F.13);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
gap> SetConjugate(coll, 8, 4, F.8*F.13);;
gap> SetConjugate(coll, 8, 7, F.8*F.13);;
gap> SetConjugate(coll, 10, 1, F.11*F.12*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.13);;
gap> SetConjugate(coll, 10, 6, F.10*F.13);;
gap> SetConjugate(coll, 10, 7, F.10*F.13);;
gap> SetConjugate(coll, 11, 1, F.8*F.9);;
gap> SetConjugate(coll, 11, 2, F.11*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 11, 5, F.11*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.13);;
gap> SetConjugate(coll, 11, 7, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.8*F.9*F.10*F.11*F.12);;
gap> SetConjugate(coll, 12, 5, F.12*F.13);;
gap> SetConjugate(coll, 12, 6, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);