Finite Group Databasebeta

← Prev (#13132560) Next (#13132562) →

C5×Q8×AGL(1,𝔽7) #13132561

order1680
automorphism?
homologyAbelianization (H1) = C23×C3×C5 · Schur multiplier (H2) = C22

Aliases

direct productC5×Q8×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C5×Q8)

Splittings

FittingC5×C7×Q8 ⋊ C2×C3
HallC5×Q8×D7 ⋊ C3
HallQ8×D7 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C2×Q8
HallC5×C7 ⋊ C2×C3×Q8
HallΓL(1,𝔽8) ⋊ C2×C5×Q8
SylowC7 ⋊ C2×C3×C5×Q8

Subgroups

subgroupG/H
centerC2×C5C22×AGL(1,𝔽7)
commutator, cosocleC2×C7C23×C3×C5
FrattiniC2C22×C5×AGL(1,𝔽7)
FittingC5×C7×Q8C2×C3
socleC2×C5×C7C23×C3
prime radicalC2×C5×AGL(1,𝔽7)C22
hypercenterC5×Q8AGL(1,𝔽7)
hypocenterC7C2×C3×C5×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×Q8

152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

175 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×SL(2,𝔽3)×AGL(1,𝔽7), Q8×D5×AGL(1,𝔽7), C5×SD16×AGL(1,𝔽7), …
maximal normal subgroup ofC5×SL(2,𝔽3)×AGL(1,𝔽7), Q8×D5×AGL(1,𝔽7), C5×SD16×AGL(1,𝔽7), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 581);