Finite Group Databasebeta

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C22×#11722398 #72252731

order3456
automorphism?
homologyAbelianization (H1) = C24 · Schur multiplier (H2) = C27

Aliases

direct productC22×#11722398

Splittings

FittingC26×C32 ⋊ D3

Subgroups

subgroupG/H
centerC23×C3C2×C3⋊S4
commutator, cosocleC2×C22⋊H(3,𝔽3)C24
FrattiniC2×C3C23×C3⋊S4
FittingC26×C32D3
socleC25×C3C2×C3⋊D3
hypercenterC24×C3C3⋊S4
hypocenterC22⋊H(3,𝔽3)C22×D4
LCSG ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C23×C3 ◁ C24×C3

8049 conjugacy classes of subgroups; only the 321 normal subgroups are shown.

normal subgroups only

This group has 8049 conjugacy classes of subgroups — too many to draw. Only its 321 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degreeμt(G) = 144

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#72213251

GAP

Past the SmallGroups library coverage — no SmallGroups id.