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C5×C3⋊(C11⋊AGL(1,𝔽7)) #20294178

order6930
automorphism?
homologyAbelianization (H1) = C2×C3×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×C3⋊(C11⋊AGL(1,𝔽7))
unambiguous semidirect productC3⋊(C5×C11⋊AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×D33)
unambiguous semidirect product(C5×D231)⋊C3

Splittings

Fitting, socleC3×C5×C7×C11 ⋊ C2×C3
derivedC3×C7×C11 ⋊ C2×C3×C5
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C2
HallC3×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC5×C7×C11 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ D11
HallC7×C11 ⋊ C3×C5×D3
HallC3×ΓL(1,𝔽8) ⋊ C5×D11
HallC5×C11 ⋊ C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D33
SylowC11 ⋊ C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C3×C5×D33

Subgroups

subgroupG/H
center, hypercenterC5C3⋊(C11⋊AGL(1,𝔽7))
commutator, hypocenterC3×C7×C11C2×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C11C2×C3
LCSG ▷ C3×C7×C11
UCSC1 ◁ C5

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

435 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×D11×C3⋊AGL(1,𝔽7), C5×AGL(1,𝔽7)×D33, C5×D3×C11⋊AGL(1,𝔽7), …
maximal normal subgroup ofC5×D11×C3⋊AGL(1,𝔽7), C5×AGL(1,𝔽7)×D33, C5×D3×C11⋊AGL(1,𝔽7), …
maximal quotient ofC5×C3⋊(C11⋊(ΓL(1,𝔽8)⋊C4)), C2×C5×C3⋊(C11⋊AGL(1,𝔽7)), C25×C3⋊(C11⋊AGL(1,𝔽7)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 65);