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D280⋊C3 #13132055

order1680
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productD280⋊C3

Splittings

FittingC5×C7×C8 ⋊ C2×C3
HallD280 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ D8
HallC5×C7 ⋊ C3×D8
HallΓL(1,𝔽8) ⋊ D40
SylowC7 ⋊ C3×D40
SylowC5 ⋊ D56⋊C3

Subgroups

subgroupG/H
centerC2C4⋊(C5⋊AGL(1,𝔽7))
commutator, cosocleC4×C5×C7C22×C3
FrattiniC4C2×C5⋊AGL(1,𝔽7)
FittingC5×C7×C8C2×C3
socleC2×C5×C7C3×D4
hypercenterC8C5⋊AGL(1,𝔽7)
hypocenterC5×C7C3×D8
LCSG ▷ C4×C5×C7 ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C2 ◁ C4 ◁ C8

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Polycyclic presentation:

2^2,3,2^2,5,7|b^2=de,d^2=e,b^a=bd,d^a=de,f^a=f^4,g^a=g^6,g^c=g^2

Conjugacy classes & character table

109 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD5×D56⋊C3, D8×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×D40, …
maximal normal subgroup ofD5×D56⋊C3, D8×C5⋊AGL(1,𝔽7), AGL(1,𝔽7)×D40, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 75);