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C7×ΓL(1,𝔽8)×#595 #23756175

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

Aliases

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

Splittings

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

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

60 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)2×#595, C3×C7×ΓL(1,𝔽8)×#595, C7×ΓL(1,𝔽8)×C3⋊#595
maximal normal subgroup ofΓL(1,𝔽8)2×#595, C3×C7×ΓL(1,𝔽8)×#595
maximal quotient ofC7×C7⋊C9×#595, C3×C7×ΓL(1,𝔽8)×#595, C7×ΓL(1,𝔽8)×C3⋊#595

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 78);