Finite Group Databasebeta

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C2×C32×D4×AGL(1,𝔽7) #72245620

order6048
automorphism?
homologyAbelianization (H1) = C24×C33 · Schur multiplier (H2) = C26×C33

Aliases

direct productC2×C32×D4×AGL(1,𝔽7)

Splittings

FittingC2×C32×C7×D4 ⋊ C2×C3
HallC2×D4×D7 ⋊ C33
HallC32×ΓL(1,𝔽8) ⋊ C22×D4
SylowC7 ⋊ C22×C33×D4

Subgroups

subgroupG/H
centerC22×C32C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C24×C33
FrattiniC2C23×C32×AGL(1,𝔽7)
FittingC2×C32×C7×D4C2×C3
socleC22×C32×C7C23×C3
hypercenterC2×C32×D4AGL(1,𝔽7)
hypocenterC7C22×C33×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22×C32 ◁ C2×C32×D4

6608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 504

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC2×D4×C3⋊D3×AGL(1,𝔽7), C32×#127×AGL(1,𝔽7), AGL(1,𝔽7)×#64149, …
maximal normal subgroup ofC2×D4×C3⋊D3×AGL(1,𝔽7), C32×#127×AGL(1,𝔽7), AGL(1,𝔽7)×#64149, …
maximal quotient ofC32×#127×AGL(1,𝔽7), C32×#120×AGL(1,𝔽7)

GAP

Past the SmallGroups library coverage — no SmallGroups id.