Finite Group Databasebeta

← Prev (#72215413) Next (#72215415) →

#72215414

order3072
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C23

Aliases

No aliases for this group.

Splittings

FittingC2×#8813 ⋊ D3

Subgroups

subgroupG/H
center, hypercenterC22#11711461
commutatorC2×C82⋊C3C2×C4
FrattiniC22×C42C2×S4
FittingC2×#8813D3
socleC24#857⋊C2
prime radicalC22×#67992C2
cosocleC22×C82⋊C3C22
hypocenterC82⋊C3C22⋊C4
LCSG ▷ C2×C82⋊C3 ▷ C82⋊C3
UCSC1 ◁ C22

1105 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1105 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.

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

Full subgroup data →

Presentation

Polycyclic presentation:

2^4,3,2^6|a^2=c,b^2=cdk,c^2=k,d^2=k,f^2=h,g^2=i,h^2=j,i^2=k,b^a=bd,c^b=ck,d^b=dk,e^a=e^2hj,e^b=ei,e^c=ehj,e^d=ehj,f^a=fgh,f^b=fh,f^e=ghjk,g^a=gk,g^b=gi,g^e=fg,h^a=hij,h^b=hj,h^e=ij,i^b=ik,i^e=hi,j^a=jk,j^e=k,k^e=jk

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72218267
maximal normal subgroup of#72218267
maximal quotient of#72215444

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