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C7×AGL(1,𝔽7)×C11⋊C5 #30633494

order16170
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×AGL(1,𝔽7)×C11⋊C5
unambiguous semidirect productD7⋊(C3×C7×C11⋊C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C7×C11⋊C5)
unambiguous semidirect product(C11×AGL(1,𝔽7))⋊(C5×C7)
unambiguous semidirect product(D7×C11⋊C5)⋊(C3×C7)
unambiguous semidirect product(C7×C11×AGL(1,𝔽7))⋊C5

Splittings

Fitting, socle, HallC72×C11 ⋊ C2×C3×C5
derivedC7×C11 ⋊ C2×C3×C5×C7
HallC7×ΓL(1,𝔽8)×C11⋊C5 ⋊ C2
HallC7×D7×C11⋊C5 ⋊ C3
HallC7×C11×AGL(1,𝔽7) ⋊ C5
HallC72×C11⋊C5 ⋊ C2×C3
HallC7×C11×ΓL(1,𝔽8) ⋊ C2×C5
HallC7×C11×D7 ⋊ C3×C5
HallC7×ΓL(1,𝔽8) ⋊ C2×C11⋊C5
HallC7×D7 ⋊ C3×C11⋊C5
SylowC72 ⋊ C2×C3×C11⋊C5
SylowC11 ⋊ C5×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7AGL(1,𝔽7)×C11⋊C5
commutator, cosocle, hypocenterC7×C11C2×C3×C5×C7
FrattiniC1G
Fitting, socleC72×C11C2×C3×C5
LCSG ▷ C7×C11
UCSC1 ◁ C7

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 462

Conjugacy classes & character table

343 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC7×AGL(1,𝔽7)×AGL(1,𝔽11), AGL(1,𝔽7)×C7⋊AGL(1,𝔽11), D7×AGL(1,𝔽7)×C11⋊C5, …
maximal normal subgroup ofC7×AGL(1,𝔽7)×AGL(1,𝔽11), AGL(1,𝔽7)×C7⋊AGL(1,𝔽11), D7×AGL(1,𝔽7)×C11⋊C5, …
maximal quotient ofC7×C11⋊C5×ΓL(1,𝔽8)⋊C4, C2×C7×AGL(1,𝔽7)×C11⋊C5

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16170, 39);