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C7×ΓL(1,𝔽8)⋊#595 #23756173

order9996
automorphism?
homologyAbelianization (H1) = C3×C4×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×ΓL(1,𝔽8)⋊#595
unambiguous semidirect productΓL(1,𝔽8)⋊(C7×#595)
unambiguous semidirect product(C7⋊#595)⋊(C3×C7)

Splittings

Fitting, socle, HallC72×C17 ⋊ C3×C4
derivedC7×C17 ⋊ C3×C4×C7
HallC7×C7⋊#595 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C4
HallC7×ΓL(1,𝔽8) ⋊ #595
SylowC72 ⋊ C3×#595
SylowC17 ⋊ C7×ΓL(1,𝔽8)⋊C4

Subgroups

subgroupG/H
center, hypercenterC7ΓL(1,𝔽8)⋊#595
commutator, hypocenterC7×C17C3×C4×C7
FrattiniC1G
Fitting, socleC72×C17C3×C4
prime radicalC7×ΓL(1,𝔽8)×D17C2
cosocleC7×D17C2×C3×C7
LCSG ▷ C7×C17
UCSC1 ◁ C7

52 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

238 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×ΓL(1,𝔽8)⋊#595, C3×C7×ΓL(1,𝔽8)⋊#595, C7×C3⋊(ΓL(1,𝔽8)⋊#595)
maximal normal subgroup ofΓL(1,𝔽8)×ΓL(1,𝔽8)⋊#595, C3×C7×ΓL(1,𝔽8)⋊#595
maximal quotient ofC7×(C7⋊C9)⋊#595, C3×C7×ΓL(1,𝔽8)⋊#595, C7×C3⋊(ΓL(1,𝔽8)⋊#595)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 76);