Finite Group Databasebeta

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C2×C4×C3⋊(C3⋊AGL(1,𝔽7)) #72245491

order3024
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C3

Aliases

direct productC2×C4×C3⋊(C3⋊AGL(1,𝔽7))

Splittings

FittingC2×C32×C4×C7 ⋊ C2×C3
derivedC32×C7 ⋊ C22×C3×C4
HallC32×ΓL(1,𝔽8) ⋊ C22×C4
SylowC7 ⋊ C2×C3×C4×C3⋊D3

Subgroups

subgroupG/H
center, hypercenterC2×C4C3⋊(C3⋊AGL(1,𝔽7))
commutator, hypocenterC32×C7C22×C3×C4
FrattiniC2C22×C3⋊(C3⋊AGL(1,𝔽7))
FittingC2×C32×C4×C7C2×C3
socleC22×C32×C7C22×C3
prime radicalC22×C3⋊(C3⋊AGL(1,𝔽7))C2
cosocleC2×C32×C7C23×C3
LCSG ▷ C32×C7
UCSC1 ◁ C2×C4

776 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 776 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 504

Conjugacy classes & character table

216 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C4×C3⋊D3×AGL(1,𝔽7), C2×D4×C3⋊(C3⋊AGL(1,𝔽7)), C2×#72245533, …
maximal normal subgroup ofC2×C4×C3⋊D3×AGL(1,𝔽7), C2×D4×C3⋊(C3⋊AGL(1,𝔽7)), C2×#72245533, …
maximal quotient ofC42×C3⋊(C3⋊AGL(1,𝔽7)), C22⋊C4×C3⋊(C3⋊AGL(1,𝔽7))

GAP

Past the SmallGroups library coverage — no SmallGroups id.