Finite Group Databasebeta

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#72248174

order6144
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C25

Aliases

No aliases for this group.

Subgroups

subgroupG/H
center, hypercenterC2#72215886
commutator#85645C22×C4
FrattiniC26C22×S4
FittingC2×#347042D3
socleC23#11713200
prime radicalC2×#72215851C2
cosocleC2×#85645C23
hypocenterC22⋊(C22⋊A4)C4×D4
LCSG ▷ #85645 ▷ C22⋊(C22⋊A4)
UCSC1 ◁ C2

At least 7251 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has at least 7251 conjugacy classes of subgroups — too many to draw. Only its 80 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:

2^5,3,2^6|b^2=d,c^a=ce,f^b=f^2,g^a=gi,g^c=gl,g^e=gl,g^f=h,h^a=hj,h^b=gh,h^c=hk,h^e=hk,h^f=gh,i^c=il,i^f=j,j^b=ij,j^c=jk,j^f=ij,k^b=kl,k^f=kl,l^f=k

Conjugacy classes & character table

96 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72248221, C4×#72215886
maximal normal subgroup ofC4×#72215886

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