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C4×C5×AGL(1,𝔽7) #11719722

order840
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C4×C5 · Schur multiplier (H2) = C2

Aliases

direct productC4×C5×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C4×C5)
unambiguous semidirect product(C4×D7)⋊(C3×C5)
unambiguous semidirect product(C4×C5×D7)⋊C3

Splittings

FittingC4×C5×C7 ⋊ C2×C3
derived, SylowC7 ⋊ C2×C3×C4×C5
HallC4×C5×D7 ⋊ C3
HallC5×ΓL(1,𝔽8) ⋊ C2×C4
HallC4×D7 ⋊ C3×C5
HallC5×C7 ⋊ C2×C3×C4
HallΓL(1,𝔽8) ⋊ C2×C4×C5

Subgroups

subgroupG/H
center, hypercenterC4×C5AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C4×C5
FrattiniC2C2×C5×AGL(1,𝔽7)
FittingC4×C5×C7C2×C3
socleC2×C5×C7C22×C3
prime radicalC2×C5×AGL(1,𝔽7)C2
cosocleC2×C7C22×C3×C5
LCSG ▷ C7
UCSC1 ◁ C4×C5

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 140

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#168×AGL(1,𝔽7), #10587838⋊C3, C5×C8×AGL(1,𝔽7), …
maximal normal subgroup of#168×AGL(1,𝔽7), #10587838⋊C3, C5×C8×AGL(1,𝔽7), …
maximal quotient ofC5×C8×AGL(1,𝔽7), C5×#1131⋊C3, C4×C5×ΓL(1,𝔽8)⋊C4, …
automorphism group ofD7×C25

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(840, 24);