Finite Group Databasebeta

← Prev (#13132951) Next (#13132953) →

C23×C5×AGL(1,𝔽7) #13132952

order1680
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C24×C3×C5 · Schur multiplier (H2) = C26

Aliases

direct productC23×C5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C23×C3×C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C24×C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C23×C5)

Splittings

Fitting, socleC23×C5×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C24×C3×C5
HallC23×C5×D7 ⋊ C3
HallC23×D7 ⋊ C3×C5
HallC5×ΓL(1,𝔽8) ⋊ C24
HallC5×C7 ⋊ C24×C3
HallΓL(1,𝔽8) ⋊ C24×C5

Subgroups

subgroupG/H
center, hypercenterC23×C5AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C24×C3×C5
FrattiniC1G
Fitting, socleC23×C5×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C23×C5

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 280

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC23×D5×AGL(1,𝔽7), C2×C5×D4×AGL(1,𝔽7), C2×C22⋊D5×AGL(1,𝔽7), …
maximal normal subgroup ofC23×D5×AGL(1,𝔽7), C2×C5×D4×AGL(1,𝔽7), C2×C22⋊D5×AGL(1,𝔽7), …
maximal quotient ofC2×C5×D4×AGL(1,𝔽7), C24×C5×AGL(1,𝔽7), C5×#41×AGL(1,𝔽7)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 972);