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C7×D17×AGL(1,𝔽7) #23756174

order9996
automorphism?
homologyAbelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D17×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C7×D17)
unambiguous semidirect product(D7×D17)⋊(C3×C7)
unambiguous semidirect product(ΓL(1,𝔽8)×D17)⋊(C2×C7)
unambiguous semidirect product(C17×AGL(1,𝔽7))⋊(C2×C7)
unambiguous semidirect product(C17⋊AGL(1,𝔽7))⋊(C2×C7)

Splittings

Fitting, socle, HallC72×C17 ⋊ C22×C3
derivedC7×C17 ⋊ C22×C3×C7
HallC7×D7×D17 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C22
HallC7×ΓL(1,𝔽8) ⋊ C2×D17
SylowC72 ⋊ C2×C3×D17
SylowC17 ⋊ C2×C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC7D17×AGL(1,𝔽7)
commutator, cosocle, hypocenterC7×C17C22×C3×C7
FrattiniC1G
Fitting, socleC72×C17C22×C3
LCSG ▷ C7×C17
UCSC1 ◁ C7

84 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

490 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofΓL(1,𝔽8)×D17×AGL(1,𝔽7), C3×C7×D17×AGL(1,𝔽7), C7×D17×C3⋊AGL(1,𝔽7), …
maximal normal subgroup ofΓL(1,𝔽8)×D17×AGL(1,𝔽7), C3×C7×D17×AGL(1,𝔽7)
maximal quotient ofC7×D17×D7⋊C9, C3×C7×D17×AGL(1,𝔽7), C7×D17×C3⋊AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 77);