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#72267211

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow#72267194 ⋊ C3

Subgroups

subgroupG/H
center, hypercenterC4#72209702
commutator, Fitting, cosocle, hypocenter#72267194C3
FrattiniC24×C4C22⋊(C22⋊A4)
socleC2#72219176
LCSG ▷ #72267194
UCSC1 ◁ C4

At least 11875 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has at least 11875 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) = 64
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(2,11,3)(4,26,34)(5,30,61)(6,32,42)(7,40,54)(12,49,20)(13,44,39)(14,28,22)(15,53,64)(16,57,24)(17,18,23)(19,50,59)(21,36,29)(25,45,48)(27,37,38)(31,62,35)(33,63,46)(41,55,43)(47,51,56)(52,58,60);(1,55,9,54,8,13,10,32)(2,34,45,35,17,5,16,36)(3,59,24,64,23,46,25,51)(4,47,31,63,30,53,29,50)(6,42,39,44,40,7,41,43)(11,60,57,38,18,20,48,28)(12,62,27,61,52,21,22,26)(14,15,58,33,37,56,49,19);(1,62,9,61,8,21,10,26)(2,53,16,63,17,47,45,50)(3,40,24,41,23,6,25,39)(4,34,29,36,30,5,31,35)(7,51,43,59,42,64,44,46)(11,37,48,58,18,14,57,49)(12,13,27,32,52,55,22,54)(15,38,19,60,56,28,33,20);(1,18,8,11)(2,3,17,23)(4,42,30,7)(5,59,34,46)(6,50,40,63)(9,57,10,48)(12,15,52,56)(13,60,55,20)(14,61,37,26)(16,24,45,25)(19,22,33,27)(21,58,62,49)(28,54,38,32)(29,44,31,43)(35,64,36,51)(39,53,41,47)

Polycyclic presentation:

3,2^12|b^2=m,c^2=m,d^2=ijm,e^2=i,f^2=l,g^2=k,i^2=m,k^2=m,l^2=m,b^a=c,c^a=bcm,c^b=cm,d^a=ehm,d^b=di,d^c=dij,e^a=dehij,e^b=eh,e^c=ehi,e^d=ehij,f^a=ghijlm,f^b=fhijk,f^c=fl,f^d=fijlm,f^e=fjkl,g^a=fghij,g^b=gl,g^c=gklm,g^d=gjklm,g^e=gikm,g^f=ghij,h^a=hj,h^c=hm,h^e=hm,h^g=hm,i^a=him,i^b=im,i^d=im,i^f=im,j^a=hm,j^b=jm,j^c=jm,j^d=jm,j^e=jm,j^f=jm,j^g=jm,k^a=hijklm,k^c=km,k^f=km,l^a=km,l^b=lm,l^c=lm,l^g=lm

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72267347

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.13);;
gap> SetPower(coll, 3, F.13);;
gap> SetPower(coll, 4, F.9*F.10*F.13);;
gap> SetPower(coll, 5, F.9);;
gap> SetPower(coll, 6, F.12);;
gap> SetPower(coll, 7, F.11);;
gap> SetPower(coll, 9, F.13);;
gap> SetPower(coll, 11, F.13);;
gap> SetPower(coll, 12, F.13);;
gap> SetConjugate(coll, 2, 1, F.3);;
gap> SetConjugate(coll, 3, 1, F.2*F.3*F.13);;
gap> SetConjugate(coll, 3, 2, F.3*F.13);;
gap> SetConjugate(coll, 4, 1, F.5*F.8*F.13);;
gap> SetConjugate(coll, 4, 2, F.4*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.4*F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.7*F.8*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 6, 3, F.6*F.12);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10*F.12*F.13);;
gap> SetConjugate(coll, 6, 5, F.6*F.10*F.11*F.12);;
gap> SetConjugate(coll, 7, 1, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.12);;
gap> SetConjugate(coll, 7, 3, F.7*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 4, F.7*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 7, 5, F.7*F.9*F.11*F.13);;
gap> SetConjugate(coll, 7, 6, F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.13);;
gap> SetConjugate(coll, 8, 5, F.8*F.13);;
gap> SetConjugate(coll, 8, 7, F.8*F.13);;
gap> SetConjugate(coll, 9, 1, F.8*F.9*F.13);;
gap> SetConjugate(coll, 9, 2, F.9*F.13);;
gap> SetConjugate(coll, 9, 4, F.9*F.13);;
gap> SetConjugate(coll, 9, 6, F.9*F.13);;
gap> SetConjugate(coll, 10, 1, F.8*F.13);;
gap> SetConjugate(coll, 10, 2, F.10*F.13);;
gap> SetConjugate(coll, 10, 3, F.10*F.13);;
gap> SetConjugate(coll, 10, 4, F.10*F.13);;
gap> SetConjugate(coll, 10, 5, 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*F.10*F.11*F.12*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 11, 6, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.11*F.13);;
gap> SetConjugate(coll, 12, 2, F.12*F.13);;
gap> SetConjugate(coll, 12, 3, F.12*F.13);;
gap> SetConjugate(coll, 12, 7, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);