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C5×ΓL(1,𝔽8)⋊AGL(1,𝔽7) #17984347

order4410
automorphism?
homologyAbelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C3

Aliases

direct productC5×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×AGL(1,𝔽7))
unambiguous semidirect product(ΓL(1,𝔽8)⋊D7)⋊(C3×C5)
unambiguous semidirect product#64424⋊(C3×C5)
unambiguous semidirect product#64425⋊(C3×C5)
unambiguous semidirect product(C5×#64424)⋊C3
unambiguous semidirect product(C5×#64425)⋊C3
unambiguous semidirect product(C5×ΓL(1,𝔽8)⋊D7)⋊C3

Splittings

Fitting, socle, HallC5×C72 ⋊ C2×C32
derived, SylowC72 ⋊ C2×C32×C5
HallC5×ΓL(1,𝔽8)2 ⋊ C2
HallC5×C7⋊D7 ⋊ C32
HallΓL(1,𝔽8)2 ⋊ C2×C5
HallC7⋊D7 ⋊ C32×C5

Subgroups

subgroupG/H
center, hypercenterC5ΓL(1,𝔽8)⋊AGL(1,𝔽7)
commutator, cosocle, hypocenterC72C2×C32×C5
FrattiniC1G
Fitting, socleC5×C72C2×C32
LCSG ▷ C72
UCSC1 ◁ C5

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 105

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC25×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C52×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C5×C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7))
maximal normal subgroup ofC25×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C52×ΓL(1,𝔽8)⋊AGL(1,𝔽7)
maximal quotient ofC25×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C52×ΓL(1,𝔽8)⋊AGL(1,𝔽7), C5×C5⋊(ΓL(1,𝔽8)⋊AGL(1,𝔽7))

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4410, 68);