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C4×Q8×AGL(1,𝔽7) #13057305

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

Aliases

direct productC4×Q8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4×Q8)

Splittings

FittingC4×C7×Q8 ⋊ C2×C3
HallC4×Q8×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4×Q8
SylowC7 ⋊ C2×C3×C4×Q8

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC4×C7×Q8C2×C3
socle, cosocleC22×C7C24×C3
prime radicalC22×AGL(1,𝔽7)C23
hypercenterC4×Q8AGL(1,𝔽7)
hypocenterC7C2×C3×C4×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×Q8

596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofQ8×AGL(1,𝔽5)×AGL(1,𝔽7), AGL(1,𝔽7)×#440, C4×SD16×AGL(1,𝔽7), …
maximal normal subgroup ofAGL(1,𝔽7)×#440, C4×SD16×AGL(1,𝔽7), C4×#41×AGL(1,𝔽7), …
maximal quotient ofQ8×AGL(1,𝔽5)×AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6594);