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ΓL(1,𝔽8)×#66776 #20297405

order7140
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×#66776
unambiguous semidirect product(C7×#66776)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C17 ⋊ C3×C4
HallC7×#66776 ⋊ C3
HallC5×C17×ΓL(1,𝔽8) ⋊ C4
HallC17×ΓL(1,𝔽8) ⋊ AGL(1,𝔽5)
HallC7×C17 ⋊ C3×AGL(1,𝔽5)
HallC5×ΓL(1,𝔽8) ⋊ #595
HallC5×C17 ⋊ C4×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×#595
SylowC17 ⋊ AGL(1,𝔽5)×ΓL(1,𝔽8)
SylowC7 ⋊ C3×#66776
SylowC5 ⋊ ΓL(1,𝔽8)×#595

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C17C3×C4
LCSG ▷ C5×C7×C17
UCSC1

48 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)×#66776, ΓL(1,𝔽8)×C3⋊#66776, ΓL(1,𝔽8)×#13133324, …
maximal normal subgroup ofC3×ΓL(1,𝔽8)×#66776, C5×ΓL(1,𝔽8)×#66776
maximal quotient ofC7⋊C9×#66776, C3×ΓL(1,𝔽8)×#66776, ΓL(1,𝔽8)×C3⋊#66776, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7140, 35);