Finite Group Databasebeta

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C32×A4×AGL(1,𝔽5) #72247476

order2160
automorphism?
homologyAbelianization (H1) = C33×C4 · Schur multiplier (H2) = C2×C33

Aliases

direct productC32×A4×AGL(1,𝔽5)

Splittings

Fitting, socleC22×C32×C5 ⋊ C3×C4
derivedC22×C5 ⋊ C33×C4
HallC22×AGL(1,𝔽5) ⋊ C33
SylowC5 ⋊ C32×C4×A4

Subgroups

subgroupG/H
center, hypercenterC32A4×AGL(1,𝔽5)
commutator, hypocenterC22×C5C33×C4
FrattiniC1G
Fitting, socleC22×C32×C5C3×C4
prime radicalC32×D5×A4C2
cosocleC22×D5C2×C33
LCSG ▷ C22×C5
UCSC1 ◁ C32

420 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 180

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×AGL(1,𝔽5)×3−1+2, C32×AGL(1,𝔽5)×C22⋊A4, C3×A42×AGL(1,𝔽5), …
maximal normal subgroup ofA4×AGL(1,𝔽5)×3−1+2, C3×D3×A4×AGL(1,𝔽5), C32×AGL(1,𝔽5)×S4, …
maximal quotient ofA4×AGL(1,𝔽5)×3−1+2, C32×AGL(1,𝔽5)×C22⋊A4, C3×A42×AGL(1,𝔽5)

GAP

Past the SmallGroups library coverage — no SmallGroups id.