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C52×C7×AGL(1,𝔽7) #20301253

order7350
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C2×C3×C52×C7 · Schur multiplier (H2) = C5

Aliases

direct productC52×C7×AGL(1,𝔽7)
unambiguous semidirect productD7⋊(C3×C52×C7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C2×C52×C7)
unambiguous semidirect product(C5×D7)⋊(C3×C5×C7)

Splittings

Fitting, socle, HallC52×C72 ⋊ C2×C3
derivedC7 ⋊ C2×C3×C52×C7
HallC52×C7×ΓL(1,𝔽8) ⋊ C2
HallC52×C7×D7 ⋊ C3
HallC7×ΓL(1,𝔽8) ⋊ C2×C52
HallC7×D7 ⋊ C3×C52
SylowC72 ⋊ C2×C3×C52

Subgroups

subgroupG/H
center, hypercenterC52×C7AGL(1,𝔽7)
commutator, hypocenterC7C2×C3×C52×C7
FrattiniC1G
Fitting, socleC52×C72C2×C3
LCSG ▷ C7
UCSC1 ◁ C52×C7

136 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1225 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC52×D7×AGL(1,𝔽7), C5×C7×D5×AGL(1,𝔽7), C5×AGL(1,𝔽7)×D35, …
maximal normal subgroup ofC52×D7×AGL(1,𝔽7), C5×C7×D5×AGL(1,𝔽7), C5×AGL(1,𝔽7)×D35, …
maximal quotient ofC52×C7×ΓL(1,𝔽8)⋊C4, C2×C52×C7×AGL(1,𝔽7), C52×C7×D7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(7350, 76);