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C3×ΓL(1,𝔽8)×D53 #20269246

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

Aliases

direct productC3×ΓL(1,𝔽8)×D53
unambiguous semidirect productC53⋊(C2×C3×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D53)⋊C32
unambiguous semidirect product(C3×C7×D53)⋊C3

Splittings

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

420 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊#6157, C3×AGL(1,𝔽7)×D53, C3×ΓL(1,𝔽8)×#6157, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)⋊#6157, C3×AGL(1,𝔽7)×D53, C3×ΓL(1,𝔽8)×#6157, …
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, 16);