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C4×D4×AGL(1,𝔽7) #13057280

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

Aliases

direct productC4×D4×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4×D4)
unambiguous semidirect product(C4×D4×D7)⋊C3

Splittings

FittingC4×C7×D4 ⋊ C2×C3
HallC4×D4×D7 ⋊ C3
HallΓL(1,𝔽8) ⋊ C2×C4×D4
SylowC7 ⋊ C2×C3×C4×D4

Subgroups

subgroupG/H
centerC2×C4C22×AGL(1,𝔽7)
commutatorC2×C7C23×C3×C4
FrattiniC22C23×AGL(1,𝔽7)
FittingC4×C7×D4C2×C3
socle, cosocleC22×C7C24×C3
prime radicalC2×D4×AGL(1,𝔽7)C2
hypercenterC4×D4AGL(1,𝔽7)
hypocenterC7C2×C3×C4×D4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C4×D4

852 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 ofD4×AGL(1,𝔽5)×AGL(1,𝔽7), C2×C4×D4×AGL(1,𝔽7), C4×AGL(1,𝔽7)×GO+(4,𝔽2), …
maximal normal subgroup ofC2×C4×D4×AGL(1,𝔽7), C4×D8×AGL(1,𝔽7), C4×#6474⋊C3, …
maximal quotient ofD4×AGL(1,𝔽5)×AGL(1,𝔽7), C2×C4×D4×AGL(1,𝔽7), C4×AGL(1,𝔽7)×GO+(4,𝔽2), …
automorphism group ofC5×Dic14, C5×D28, C2×C5×Dic7

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6569);