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C5×C7×C5⋊AGL(1,𝔽7) #20301261

order7350
isoclinismisoclinic class of C5⋊AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C5×C7 · Schur multiplier (H2) = C1

Aliases

direct productC5×C7×C5⋊AGL(1,𝔽7)
unambiguous semidirect productC5⋊(C5×C7×AGL(1,𝔽7))
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C7×D5)
unambiguous semidirect productD35⋊(C3×C5×C7)
unambiguous semidirect product(C5×D35)⋊(C3×C7)

Splittings

Fitting, socle, HallC52×C72 ⋊ C2×C3
derivedC5×C7 ⋊ C2×C3×C5×C7
HallC52×C7×ΓL(1,𝔽8) ⋊ C2
HallC5×C7×D35 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C5×D5
SylowC72 ⋊ C3×C5×D5
SylowC52 ⋊ C7×AGL(1,𝔽7)

Subgroups

subgroupG/H
center, hypercenterC5×C7C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C2×C3×C5×C7
FrattiniC1G
Fitting, socleC52×C72C2×C3
LCSG ▷ C5×C7
UCSC1 ◁ C5×C7

88 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

595 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC5×C7×D5×AGL(1,𝔽7), C5×D7×C5⋊AGL(1,𝔽7), C5×(C5⋊AGL(1,𝔽7))⋊D7, …
maximal normal subgroup ofC5×C7×D5×AGL(1,𝔽7), C5×D7×C5⋊AGL(1,𝔽7), C5×(C5⋊AGL(1,𝔽7))⋊D7, …
maximal quotient ofC5×C7×ΓL(1,𝔽8)⋊Dic5, C2×C5×C7×C5⋊AGL(1,𝔽7), C5×C7×C5⋊(D7⋊C9), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 84);