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C2×#41×AGL(1,𝔽7) #13062151

order1344
automorphism?
homologyAbelianization (H1) = C25×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×#41×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×#41)

Splittings

FittingC2×C7×#41 ⋊ C2×C3
HallC2×D7×#41 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×#41
SylowC7 ⋊ C22×C3×#41

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C25×C3
FrattiniC2C24×AGL(1,𝔽7)
FittingC2×C7×#41C2×C3
socleC22×C7C24×C3
hypercenterC2×#41AGL(1,𝔽7)
hypocenterC7C22×C3×#41
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×#41

1780 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×#41×AGL(1,𝔽7), C2×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,𝔽7)×#417, …
maximal normal subgroup ofC2×AGL(1,ℤ/8)×AGL(1,𝔽7), AGL(1,𝔽7)×#417, C4×#41×AGL(1,𝔽7), …
maximal quotient ofC2×C4×D4×AGL(1,𝔽7), D5×#41×AGL(1,𝔽7), C4×#41×AGL(1,𝔽7), …
automorphism group ofC2×C7⋊C16, #6386

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 11440);