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C11×C3⋊(C5⋊AGL(1,𝔽7)) #20294181

order6930
automorphism?
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

direct productC11×C3⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect productC3⋊(C11×C5⋊AGL(1,𝔽7))
unambiguous semidirect productC5⋊(C11×C3⋊AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C11×D15)
unambiguous semidirect productD105⋊(C3×C11)
unambiguous semidirect product(C11×D105)⋊C3

Splittings

Fitting, socleC3×C5×C7×C11 ⋊ C2×C3
derivedC3×C5×C7 ⋊ C2×C3×C11
HallC3×C5×C11×ΓL(1,𝔽8) ⋊ C2
HallC3×C11×ΓL(1,𝔽8) ⋊ D5
HallC5×C7×C11 ⋊ C3×D3
HallC3×C5×ΓL(1,𝔽8) ⋊ C2×C11
HallC7×C11 ⋊ C3×D15
HallC3×ΓL(1,𝔽8) ⋊ D5×C11
HallC5×C11 ⋊ C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C3×D3×C11
SylowC7 ⋊ C3×C11×D15
SylowC5 ⋊ C11×C3⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC11C3⋊(C5⋊AGL(1,𝔽7))
commutator, hypocenterC3×C5×C7C2×C3×C11
FrattiniC1G
Fitting, socleC3×C5×C7×C11C2×C3
LCSG ▷ C3×C5×C7
UCSC1 ◁ C11

72 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

462 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD11×C3⋊(C5⋊AGL(1,𝔽7)), C11×D15×AGL(1,𝔽7), (C11⋊AGL(1,𝔽7))⋊D15, …
maximal normal subgroup ofD11×C3⋊(C5⋊AGL(1,𝔽7)), C11×D15×AGL(1,𝔽7), (C11⋊AGL(1,𝔽7))⋊D15, …
maximal quotient ofC11×C3⋊(ΓL(1,𝔽8)⋊Dic5), C2×C11×C3⋊(C5⋊AGL(1,𝔽7)), C11×C3⋊(ΓL(1,𝔽8)⋊D25), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(6930, 68);