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C2×ΓL(1,𝔽8)×D41 #16812239

order3444
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC2×ΓL(1,𝔽8)×D41
unambiguous semidirect productC41⋊(C22×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C41)⋊(C2×ΓL(1,𝔽8))
unambiguous semidirect product(C2×C7×D41)⋊C3

Splittings

Fitting, socleC2×C7×C41 ⋊ C2×C3
derived, HallC7×C41 ⋊ C22×C3
HallC2×C7×D41 ⋊ C3
HallΓL(1,𝔽8)×C41 ⋊ C22
SylowC41 ⋊ C22×ΓL(1,𝔽8)
SylowC7 ⋊ C2×C3×D41

Subgroups

subgroupG/H
center, hypercenterC2ΓL(1,𝔽8)×D41
commutator, cosocle, hypocenterC7×C41C22×C3
FrattiniC1G
Fitting, socleC2×C7×C41C2×C3
LCSG ▷ C7×C41
UCSC1 ◁ C2

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 574

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×ΓL(1,𝔽8)×D41⋊C5, D5×ΓL(1,𝔽8)×D41, C2×C5×ΓL(1,𝔽8)×D41, …
maximal normal subgroup ofC2×ΓL(1,𝔽8)×D41⋊C5, C2×C5×ΓL(1,𝔽8)×D41, C2×C7×ΓL(1,𝔽8)×D41, …
maximal quotient ofD5×ΓL(1,𝔽8)×D41, C2×C5×ΓL(1,𝔽8)×D41, C2×ΓL(1,𝔽8)×D205, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3444, 19);