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C2×C7×C17×AGL(1,𝔽7) #23756211

order9996
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C3×C7×C17 · Schur multiplier (H2) = C2

Aliases

direct productC2×C7×C17×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×C7×C17)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C7×C17)
unambiguous semidirect product(C2×D7)⋊(C3×C7×C17)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C7×C17)
unambiguous semidirect product(C2×D7×C17)⋊(C3×C7)

Splittings

Fitting, socleC2×C72×C17 ⋊ C2×C3
derivedC7 ⋊ C22×C3×C7×C17
HallC2×C7×D7×C17 ⋊ C3
HallC7×C17×ΓL(1,𝔽8) ⋊ C22
HallC72×C17 ⋊ C22×C3
HallC2×C7×D7 ⋊ C3×C17
HallC7×ΓL(1,𝔽8) ⋊ C22×C17
SylowC72 ⋊ C22×C3×C17

Subgroups

subgroupG/H
center, hypercenterC2×C7×C17AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C3×C7×C17
FrattiniC1G
Fitting, socleC2×C72×C17C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C7×C17

84 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1666 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C17×ΓL(1,𝔽8)×AGL(1,𝔽7), D3×C7×C17×AGL(1,𝔽7), C2×C3×C7×C17×AGL(1,𝔽7), …
maximal normal subgroup ofC2×C17×ΓL(1,𝔽8)×AGL(1,𝔽7), C2×C3×C7×C17×AGL(1,𝔽7)
maximal quotient ofC2×C7×C17×D7⋊C9, D3×C7×C17×AGL(1,𝔽7), C2×C3×C7×C17×AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9996, 114);