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C3×ΓL(1,𝔽8)⋊D53 #20269245

order6678
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×ΓL(1,𝔽8)⋊D53
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×D53)
unambiguous semidirect productC53⋊(C3×AGL(1,𝔽7))
unambiguous semidirect productD371⋊C32
unambiguous semidirect product(C3×D371)⋊C3

Splittings

Fitting, socleC3×C7×C53 ⋊ C2×C3
derived, HallC7×C53 ⋊ C2×C32
HallC3×ΓL(1,𝔽8)×C53 ⋊ C2
HallD371 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ D53
SylowC53 ⋊ C3×AGL(1,𝔽7)
SylowC7 ⋊ C32×D53

Subgroups

subgroupG/H
center, hypercenterC3ΓL(1,𝔽8)⋊D53
commutator, hypocenterC7×C53C2×C32
FrattiniC1G
Fitting, socleC3×C7×C53C2×C3
LCSG ▷ C7×C53
UCSC1 ◁ C3

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

411 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×D53, D3×ΓL(1,𝔽8)⋊D53, (C3⋊AGL(1,𝔽7))⋊D53, …
maximal normal subgroup ofC3×AGL(1,𝔽7)×D53, D3×ΓL(1,𝔽8)⋊D53, (C3⋊AGL(1,𝔽7))⋊D53, …
maximal quotient ofC3×ΓL(1,𝔽8)⋊Dic53, C2×C3×ΓL(1,𝔽8)⋊D53, C3×C5×ΓL(1,𝔽8)⋊D53, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6678, 15);