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C3×C5⋊(C11⋊AGL(1,𝔽7)) #20294174

order6930
automorphism?
homologyAbelianization (H1) = C2×C32 · Schur multiplier (H2) = C3

Aliases

direct productC3×C5⋊(C11⋊AGL(1,𝔽7))
unambiguous semidirect productC5⋊(C3×C11⋊AGL(1,𝔽7))
unambiguous semidirect productC11⋊(C3×C5⋊AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C3×D55)
unambiguous semidirect productD385⋊C32
unambiguous semidirect product(C3×D385)⋊C3

Splittings

Fitting, socleC3×C5×C7×C11 ⋊ C2×C3
derived, HallC5×C7×C11 ⋊ C2×C32
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C2
HallD385 ⋊ C32
HallC3×C11×ΓL(1,𝔽8) ⋊ D5
HallC3×C5×ΓL(1,𝔽8) ⋊ D11
HallC7×C11 ⋊ C32×D5
HallC3×ΓL(1,𝔽8) ⋊ D55
HallC5×C11 ⋊ C3×AGL(1,𝔽7)
HallC5×C7 ⋊ C32×D11
SylowC11 ⋊ C3×C5⋊AGL(1,𝔽7)
SylowC7 ⋊ C32×D55
SylowC5 ⋊ C3×C11⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC3C5⋊(C11⋊AGL(1,𝔽7))
commutator, hypocenterC5×C7×C11C2×C32
FrattiniC1G
Fitting, socleC3×C5×C7×C11C2×C3
LCSG ▷ C5×C7×C11
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

426 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×D11×C5⋊AGL(1,𝔽7), C3×AGL(1,𝔽7)×D55, C3×D5×C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofC3×D11×C5⋊AGL(1,𝔽7), C3×AGL(1,𝔽7)×D55, C3×D5×C11⋊AGL(1,𝔽7), …
maximal quotient ofC3×C11⋊(ΓL(1,𝔽8)⋊Dic5), C2×C3×C5⋊(C11⋊AGL(1,𝔽7)), C3×C11⋊(ΓL(1,𝔽8)⋊D25), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 61);