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ΓL(1,𝔽16)×C41⋊C5 #27134226

order12300
automorphism?
homologyAbelianization (H1) = C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽16)×C41⋊C5
unambiguous semidirect productC3⋊(AGL(1,𝔽5)×C41⋊C5)
unambiguous semidirect product(C41×ΓL(1,𝔽16))⋊C5

Splittings

Fitting, socle, derivedC3×C5×C41 ⋊ C4×C5
HallC3×C5×C41⋊C5 ⋊ C4
HallC5×C41⋊C5 ⋊ Dic3
HallC3×C41 ⋊ C5×AGL(1,𝔽5)
SylowC41 ⋊ C5×ΓL(1,𝔽16)
SylowC3 ⋊ AGL(1,𝔽5)×C41⋊C5

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC3×C5×C41C4×C5
prime radicalC3×D5×C41⋊C5C2
cosocleC3×D5×C41C2×C5
LCSG ▷ C3×C5×C41
UCSC1

52 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC9⋊AGL(1,𝔽5)×C41⋊C5, C3×ΓL(1,𝔽16)×C41⋊C5, C3⋊ΓL(1,𝔽16)×C41⋊C5
maximal normal subgroup ofC3×ΓL(1,𝔽16)×C41⋊C5
maximal quotient ofC9⋊AGL(1,𝔽5)×C41⋊C5, C3×ΓL(1,𝔽16)×C41⋊C5, C3⋊ΓL(1,𝔽16)×C41⋊C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12300, 88);