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ΓL(1,𝔽8)×D205 #22575574

order8610
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×D205
unambiguous semidirect productC5⋊(ΓL(1,𝔽8)×D41)
unambiguous semidirect productC41⋊(D5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×D205)⋊C3

Splittings

Fitting, socle, derived, HallC5×C7×C41 ⋊ C2×C3
HallC5×ΓL(1,𝔽8)×C41 ⋊ C2
HallC7×D205 ⋊ C3
HallΓL(1,𝔽8)×C41 ⋊ D5
HallC7×C41 ⋊ C3×D5
HallC5×C41 ⋊ C2×ΓL(1,𝔽8)
HallC5×ΓL(1,𝔽8) ⋊ D41
SylowC41 ⋊ D5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×D41
SylowC7 ⋊ C3×D205
SylowC5 ⋊ ΓL(1,𝔽8)×D41

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC5×C7×C41C2×C3
LCSG ▷ C5×C7×C41
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

520 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊#11717961, ΓL(1,𝔽8)⋊#11717962, ΓL(1,𝔽8)×#11717961, …
maximal normal subgroup ofΓL(1,𝔽8)⋊#11717961, ΓL(1,𝔽8)⋊#11717962, ΓL(1,𝔽8)×#11717961, …
maximal quotient ofΓL(1,𝔽8)×Dic205, C2×ΓL(1,𝔽8)×D205, C7⋊C9×D205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(8610, 31);