Finite Group Databasebeta

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C2×C32×C4×AGL(1,𝔽7) #72245490

order3024
automorphism?
homologyAbelianization (H1) = C22×C33×C4 · Schur multiplier (H2) = C23×C33

Aliases

direct productC2×C32×C4×AGL(1,𝔽7)

Splittings

FittingC2×C32×C4×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C33×C4
HallC32×ΓL(1,𝔽8) ⋊ C22×C4
HallC2×C4×D7 ⋊ C33

Subgroups

subgroupG/H
center, hypercenterC2×C32×C4AGL(1,𝔽7)
commutator, hypocenterC7C22×C33×C4
FrattiniC2C22×C32×AGL(1,𝔽7)
FittingC2×C32×C4×C7C2×C3
socleC22×C32×C7C22×C3
prime radicalC22×C32×AGL(1,𝔽7)C2
cosocleC2×C7C23×C33
LCSG ▷ C7
UCSC1 ◁ C2×C32×C4

1512 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 504

Conjugacy classes & character table

504 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C4×C3⋊D3×AGL(1,𝔽7), C2×C32×D4×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C4⋊(C3⋊D3), …
maximal normal subgroup ofC2×C4×C3⋊D3×AGL(1,𝔽7), C2×C32×D4×AGL(1,𝔽7), C2×AGL(1,𝔽7)×C4⋊(C3⋊D3), …
maximal quotient ofC32×C42×AGL(1,𝔽7), C32×C22⋊C4×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.