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C2×C3×C5×AGL(1,𝔽7) #11929523

order1260
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C22×C32×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC2×C3×C5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C2×C32×C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C22×C3×C5)
unambiguous semidirect product(C2×D7)⋊(C32×C5)
unambiguous semidirect product(C2×ΓL(1,𝔽8))⋊(C2×C3×C5)
unambiguous semidirect product(C2×C3×D7)⋊(C3×C5)
unambiguous semidirect product(C2×C5×D7)⋊C32
unambiguous semidirect product(C2×C3×C5×D7)⋊C3

Splittings

Fitting, socleC2×C3×C5×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C22×C32×C5
HallC3×C5×ΓL(1,𝔽8) ⋊ C22
HallC2×C5×D7 ⋊ C32
HallC3×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7 ⋊ C22×C32
HallC2×D7 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC2×C3×C5AGL(1,𝔽7)
commutator, cosocle, hypocenterC7C22×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C3×C5

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 210

Conjugacy classes & character table

210 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC3×AGL(1,𝔽7)×AGL(1,𝔽11), C2×C3×ΓL(1,𝔽8)⋊AGL(1,𝔽11), C2×C3×AGL(1,𝔽7)×C11⋊C5, …
maximal normal subgroup ofC2×C3×C5×C11×AGL(1,𝔽7), C2×C3×C5×C13×AGL(1,𝔽7), C2×C3×C5×C17×AGL(1,𝔽7), …
maximal quotient ofC3×AGL(1,𝔽7)×AGL(1,𝔽11), C2×C3×ΓL(1,𝔽8)⋊AGL(1,𝔽11), C2×C3×AGL(1,𝔽7)×C11⋊C5, …
automorphism group ofD7×C31

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 75);