Finite Group Databasebeta

← Prev (#13057235) Next (#13057237) →

C2×C4⋊C4×AGL(1,𝔽7) #13057236

order1344
automorphism?
homologyAbelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26

Aliases

direct productC2×C4⋊C4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C3×C4⋊C4)
unambiguous semidirect product(C2×D7×C4⋊C4)⋊C3

Splittings

FittingC2×C7×C4⋊C4 ⋊ C2×C3
HallC2×D7×C4⋊C4 ⋊ C3
HallΓL(1,𝔽8) ⋊ C22×C4⋊C4
SylowC7 ⋊ C22×C3×C4⋊C4

Subgroups

subgroupG/H
centerC23C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC2×C7×C4⋊C4C2×C3
socleC23×C7C23×C3
prime radicalC23×AGL(1,𝔽7)C22
cosocleC22×C7C24×C3
hypercenterC2×C4⋊C4AGL(1,𝔽7)
hypocenterC7C22×C3×C4⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C4⋊C4

836 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 ofC2×AGL(1,𝔽7)×C4⋊AGL(1,𝔽5), C2×C4×D4×AGL(1,𝔽7), AGL(1,𝔽7)×#428, …
maximal normal subgroup ofC2×C4×D4×AGL(1,𝔽7), AGL(1,𝔽7)×#428, AGL(1,𝔽7)×#385, …
maximal quotient ofC2×AGL(1,𝔽7)×C4⋊AGL(1,𝔽5), AGL(1,𝔽7)×#428, AGL(1,𝔽7)×#385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6525);