Finite Group Databasebeta

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#72227576

order3072
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

Fitting, Sylow#72210362 ⋊ C3
derivedC22×H(3,𝔽4) ⋊ C22×C3

Subgroups

subgroupG/H
centerC2#72209719
commutator, cosocle, hypocenterC22×H(3,𝔽4)C22×C3
FrattiniC23×Q8C22×A4
Fitting#72210362C3
socleC23C2×B(4,𝔽2)⋊C3
hypercenterC22#11709314
LCSG ▷ C22×H(3,𝔽4)
UCSC1 ◁ C2 ◁ C22

4392 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 4392 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 24
transitive degreeμt(G) = 96

A minimal faithful permutation representation — 2 generators on 24 points:

(1,2)(3,5,7,6,4,8)(10,14)(11,19,12)(15,24,16,17,23,18)(20,21,22);(2,8)(5,6)(9,23,21,14,13,24,19,10)(11,16,22,17,20,18,12,15)

Polycyclic presentation:

2^2,3,2^8|f^2=k,g^2=k,h^2=k,i^2=k,b^a=bk,d^a=df,d^b=dfhi,d^c=e,e^a=eg,e^b=eghk,e^c=de,f^a=fk,f^c=g,f^d=fk,f^e=fj,g^a=gk,g^c=fgj,g^d=gjk,g^e=gk,g^f=gk,h^a=hk,h^c=ijk,h^d=hjk,h^e=hk,h^f=hk,i^c=hij,i^d=ij,i^e=ijk,i^f=ik,i^g=ik,i^h=ik,j^a=jk

Conjugacy classes & character table

54 conjugacy classes — view the full character table.

Relationships

maximal quotient of#72227577, #72267886

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