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C2×M16×AGL(1,𝔽7) #13057477

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C25

Aliases

direct productC2×M16×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×M16)
unambiguous semidirect product(C2×D7×M16)⋊C3

Splittings

FittingC2×C7×M16 ⋊ C2×C3
HallC2×D7×M16 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×M16
SylowC7 ⋊ C22×C3×M16

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC4C23×AGL(1,𝔽7)
FittingC2×C7×M16C2×C3
socleC22×C7C22×C3×C4
prime radicalC23×AGL(1,𝔽7)C4
cosocleC4×C7C24×C3
hypercenterC2×M16AGL(1,𝔽7)
hypocenterC7C22×C3×M16
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×M16

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 112

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×M16×AGL(1,𝔽7), AGL(1,𝔽7)×D5⋊M16, C4×M16×AGL(1,𝔽7), …
maximal normal subgroup ofC4×M16×AGL(1,𝔽7), C2×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,𝔽7)×#428, …
maximal quotient ofD5×M16×AGL(1,𝔽7), AGL(1,𝔽7)×D5⋊M16, C4×M16×AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6766);