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C2×C4⋊(C5⋊AGL(1,𝔽7)) #13132577

order1680
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC2×C4⋊(C5⋊AGL(1,𝔽7))
unambiguous semidirect product(C2×D140)⋊C3

Splittings

FittingC2×C4×C5×C7 ⋊ C2×C3
HallC2×D140 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×D4
HallC5×C7 ⋊ C2×C3×D4
HallΓL(1,𝔽8) ⋊ C2×D20
SylowC7 ⋊ C2×C3×D20
SylowC5 ⋊ C2×C4⋊AGL(1,𝔽7)

Subgroups

subgroupG/H
centerC22C2×C5⋊AGL(1,𝔽7)
commutator, cosocleC2×C5×C7C23×C3
FrattiniC2C22×C5⋊AGL(1,𝔽7)
FittingC2×C4×C5×C7C2×C3
socleC22×C5×C7C22×C3
hypercenterC2×C4C5⋊AGL(1,𝔽7)
hypocenterC5×C7C2×C3×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22 ◁ C2×C4

216 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×D5×C4⋊AGL(1,𝔽7), C2×D4×C5⋊AGL(1,𝔽7), C2×D20×AGL(1,𝔽7), …
maximal normal subgroup ofC2×D5×C4⋊AGL(1,𝔽7), C2×D4×C5⋊AGL(1,𝔽7), C2×D20×AGL(1,𝔽7), …
maximal quotient of#72218044, #72218376

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 597);